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Intermediate Algebra - 2e, 2017a

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<strong>Intermediate</strong> <strong>Algebra</strong> <strong>2e</strong><br />

SENIOR CONTRIBUTING AUTHORS<br />

LYNN MARECEK, SANTA ANA COLLEGE<br />

ANDREA HONEYCUTT MATHIS, NORTHEAST MISSISSIPPI COMMUNITY COLLEGE


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n n.<br />

<br />

15 ÷ 3 5·3.<br />

<br />

m <br />

<br />

<br />

<br />

<br />

<br />

3.<br />

0.<br />

3.<br />

0.


a b c d <br />

a<br />

<br />

<br />

<br />

<br />

<br />

b<br />

<br />

5+6+2+5=18<br />

<br />

<br />

<br />

c<br />

<br />

<br />

<br />

d<br />

<br />

<br />

<br />

<br />

a b c d e <br />

<br />

a b c d e <br />

<br />

<br />

<br />

8·9=72, 72 = 8 · 9, <br />

1 · 72, 2 · 36, 3 · 24, 4 · 18, 6 · 12. <br />

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.<br />

<br />

a · b = m,


2·2·2·2·3 <br />

<br />

6·8, <br />

<br />

80.<br />

60.


(2·2·3) (2·3·3) (2·2·3·3).<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

x, y, a, b, c.


a + b a b a b<br />

a − b a b a b<br />

<br />

a · b, ab, (a)(b),<br />

(a)b, a(b)<br />

a b<br />

a b<br />

a ÷ b, a/b, a b , ba a <br />

b<br />

a b;<br />

a b <br />

<br />

<br />

<br />

a = b <br />

<br />

<br />

<br />

<br />

a < b a > b <br />

<br />

a < b is equivalent to b > a. For example, 7 < 11 is equivalent to 11 > 7.<br />

a > b is equivalent to b < a. For example, 17 > 4 is equivalent to 4 < 17.<br />

<br />

<br />

a ≠ b<br />

a < b<br />

a ≤ b<br />

a > b<br />

a ≥ b


Parentheses ()<br />

Brackets []<br />

Braces {}<br />

<br />

<br />

8(14 − 8) 21 − 3[2 + 4(9 − 8)] 24 ÷ ⎧ ⎩13−2[1(6−5)+4] ⎨ ⎫ ⎭ ⎬<br />

<br />

<br />

<br />

<br />

<br />

Expression Words English Phrase<br />

3 + 5 3 plus 5 the sum of three and five<br />

n −1 n minus one the difference of n and one<br />

6 · 7 6 times 7 the product of six and seven<br />

x<br />

y x divided by y the quotient of x and y<br />

<br />

<br />

<br />

<br />

<br />

Equation English Sentence<br />

3 + 5 = 8 The sum of three and five is equal to eight.<br />

n −1=14 n minus one equals fourteen.<br />

6 · 7 = 42 The product of six and seven is equal to forty-two.<br />

x =53<br />

x is equal to fifty-three.<br />

y +9=2y −3 y plus nine is equal to two y minus three.<br />

2·2·2·2·2·2·2·2·2. <br />

2·2·2 2 3 2·2·2·2·2·2·2·2·2 2 9 . <br />

2 3 , <br />

<br />

<br />

2 3 2·2·2 <br />

a n


a n n th <br />

a n “a n th <br />

a 2<br />

a 3<br />

“a squared”<br />

“a cubed”<br />

a 2 a 3 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

4·2+1 <br />

4·2 <br />

<br />

4·2+1<br />

8+1<br />

9<br />

<br />

<br />

<br />

<br />

<br />

4+3·7. 4+3 <br />

3·7


Parentheses<br />

Exponents<br />

Multiplication Division<br />

Addition Subtraction<br />

Please<br />

Excuse<br />

My Dear<br />

Aunt Sally<br />

<br />

<br />

<br />

<br />

<br />

18 ÷ 6 + 4(5−2).<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

30 ÷ 5 + 10(3 − 2).<br />

70 ÷ 10 + 4(6 − 2).


5+2 3 +3 ⎡ ⎣6−3(4−2) ⎤ ⎦.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

9 +5 3 − ⎡ ⎣ ⎡ 4(9 + 3) ⎤ ⎦ ⎤ .<br />

<br />

7 2 − 2 ⎡ ⎣ ⎡ 4(5 + 1) ⎤ ⎦ ⎤ .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

x =4: a x 2 b 3 x c 2x 2 +3x +8.<br />

a


<br />

<br />

c<br />

<br />

<br />

x = 3, a x 2 b 4 x c 3x 2 + 4x + 1.<br />

<br />

x = 6, a x 3 b 2 x c 6x 2 − 4x − 7.<br />

<br />

<br />

<br />

<br />

7, y, 5x 2 ,9a, b 5 .<br />

<br />

<br />

<br />

3x x,<br />

x =1·x.<br />

<br />

<br />

<br />

5x 7 n 2 4 3x 9n 2


7 4 <br />

5x 3x <br />

n 2 9n 2 <br />

<br />

<br />

<br />

<br />

Simplify.<br />

4x +7x + x<br />

Add the coefficients.<br />

12x<br />

<br />

<br />

2x 2 +3x +7+x 2 +4x +5.<br />

<br />

<br />

3x 2 + 7x + 9 + 7x 2 + 9x + 8.<br />

<br />

4y 2 + 5y + 2 + 8y 2 + 4y + 5.


a <br />

<br />

<br />

<br />

<br />

b<br />

<br />

<br />

<br />

<br />

<br />

a b<br />

<br />

<br />

<br />

<br />

<br />

a + b<br />

a − b<br />

a · b, ab, a(b), (a)(b)<br />

2a<br />

a ÷ b, a/b, a b , ba<br />

<br />

<br />

<br />

<br />

<br />

a 14x b 8y 2 c y d 49x 2<br />

<br />

a <br />

<br />

b <br />

c


twelve more than y<br />

twelve added to y<br />

y +12<br />

d <br />

seven less than 49x 2<br />

seven subtracted from 49x 2<br />

49x 2 −7<br />

<br />

<br />

a 14x 2 b 12x c z<br />

d 8x<br />

<br />

<br />

a 17y 2 b 7 c d <br />

<br />

<br />

<br />

a b <br />

<br />

<br />

a (x + y). <br />

<br />

eight times the sum of x and y<br />

8(x + y)<br />

b <br />

<br />

<br />

<br />

a <br />

b <br />

<br />

<br />

a <br />

b


2 3 −12÷(9−5) 3 2 −18÷(11−5) 2+8(6+1)<br />

4+6(3+6) 20 ÷ 4 + 6(5−1) 33 ÷ 3 + 4(7−2)<br />

3(1+9·6)−4 2 5(2 + 8 · 4) − 7 2 2 ⎡ ⎣1+3(10 − 2) ⎤ ⎦<br />

5 ⎡ ⎣2+4(3−2) ⎤ ⎦<br />

8+2 ⎡ ⎣7−2(5−3) ⎤ ⎦ −3 2 10 + 3 ⎡ ⎣6−2(4−2) ⎤ ⎦ −2 4<br />

<br />

<br />

x =2,<br />

x =3,<br />

a x 6<br />

b 4 x<br />

c 2x 2 +3x −7<br />

a x 5<br />

b 5 x<br />

c 3x 2 −4x −8<br />

x =4,y =1<br />

x 2 +3xy −7y 2<br />

x =3,y =2<br />

x = 10, y =7<br />

a =3,b =8<br />

6x 2 +3xy −9y 2 (x − y) 2 a 2 + b 2<br />

<br />

<br />

7x +2+3x +4<br />

8y +5+2y −4<br />

10a +7+5a −2+7a −4<br />

7c +4+6c −3+9c −1


3x 2 +12x +11+14x 2 +8x +5<br />

5b 2 +9b +10+2b 2 +3b −4<br />

<br />

<br />

<br />

a 5x 2 6xy<br />

b 6y 2 5x<br />

c y 2<br />

d 6x 81x 2<br />

<br />

a 4ab 2 3a 2 b<br />

b 4y 2 5x<br />

c m<br />

d 9x 121x 2<br />

<br />

a y <br />

b y <br />

<br />

a 17x 2 5xy<br />

b 8y 3 3x<br />

c a 2 <br />

d 11b 100b 2<br />

<br />

a 3x 2 y 7xy 2<br />

b 6xy 2 4z<br />

c 3x 2<br />

d 7x 2 63x 3<br />

<br />

a y <br />

b y <br />

<br />

a 3x y<br />

b 3x y<br />

<br />

a 4x 2 5x<br />

b 4x 2 5x<br />

<br />

<br />

<br />

<br />

<br />

<br />

m <br />

<br />

<br />

<br />

<br />

<br />

<br />

$5 $10 <br />

<br />

t <br />

<br />

<br />

<br />

<br />

<br />

8a 2 +4a +9−a 2 −1.


a <br />

b


2 −2 <br />

2 −2, −2 2.<br />

<br />

<br />

<br />

<br />

<br />

−3 <br />

<br />

−a means the opposite of the number a<br />

The notation −a is read as “the opposite of a.”<br />

−3


n |n| |n| ≥0 <br />

<br />

<br />

−5 is 5 units away from 0, so |−5| =5.<br />

5 is 5 units away from 0, so |5| =5.<br />

<br />

−5 <br />

<br />

<br />

<br />

<br />

<br />

, = <br />

a |−5|__ − |−5| b 8__ − |−8| c −9__ − |−9| d −(−16)__|−16|.<br />

a<br />

<br />

|−5|__ − |−5|<br />

5__ − 5<br />

<br />

5> −5<br />

|−5| >−|−5|<br />

b<br />

8__ − |−8|<br />

8__ − 8<br />

<br />

8> −8<br />

8> −|−8|<br />

c


−9__ − |−9|<br />

<br />

<br />

−9__−9<br />

−9 = −9<br />

−9 = − |−9|<br />

d<br />

−(−16)__|−16|<br />

<br />

16__16<br />

16 = 16<br />

−(−16) = |−16|<br />

< , > , = <br />

a −9__ − |−9| b 2__ − |−2| c −8__|−8| d −(−9)__|−9|.<br />

< , > , = <br />

a 7__ − |−7| b −(−10)__|−10| c |−4|__ − |−4| d −1__|−1|.<br />

<br />

<br />

<br />

Parentheses () Braces {}<br />

Brackets [] Absolute value ||<br />

<br />

<br />

24 − |19 − 3(6−2)|.


24 − |19 − 3(6−2)|<br />

<br />

24 − |19 − 3(4)|<br />

24 − |19 − 12|<br />

24 − |7|<br />

24 − 7<br />

17<br />

19 − |11 − 4(3 − 1)|.<br />

9 − |8 − 4(7 − 5)|.<br />

<br />

<br />

Counting numbers<br />

Whole numbers<br />

1, 2, 3 …<br />

0, 1, 2, 3….<br />

<br />

… −3, −2, −1, 0, 1, 2, 3…<br />

<br />

<br />

<br />

… −3, −2, −1, 0, 1, 2, 3…,<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

5+3 −5+(−3) −5+3 5+(−3)<br />

5+3,


−5 + (−3), <br />

<br />

<br />

−5+3 5+(−3).<br />

<br />

<br />

<br />

a −1 + (−4) b −1 + 5 c 1+(−5).<br />

a<br />

<br />

<br />

b<br />

<br />

c


a −2 + (−4) b −2 + 4 c 2 + (−4).<br />

<br />

a −2 + (−5) b −2 + 5 c 2 + (−5).<br />

“5 − 3” <br />

<br />

<br />

5−3 −5−(−3) −5−3 5−(−3)<br />

5−3, <br />

−5 − (−3), <br />

<br />

<br />

<br />

<br />

<br />

−5−3 5−(−3).


a 3−1 b −3 − (−1) c −3−1 d 3−(−1).<br />

a<br />

<br />

<br />

b<br />

<br />

c<br />

<br />

d<br />

<br />

<br />

a 6 − 4 b −6 − (−4) c −6 − 4 d 6 − (−4).<br />

<br />

a 7 − 4 b −7 − (−4) c −7 − 4 d 7 − (−4).


−3 − 1 <br />

−3 + (−1) 3−(−1) 3+1. <br />

<br />

<br />

a − b = a + (−b)<br />

<br />

<br />

a 13 − 8 13 + (−8) b −17−9 −17 + (−9) c 9 − (−15) 9+15 d −7 − (−4) <br />

−7+4.<br />

a<br />

<br />

<br />

13 − 8 and 13 + (−8)<br />

5 5<br />

b<br />

<br />

−17 − 9 and −17 + (−9)<br />

−26<br />

−26<br />

c<br />

<br />

9−(−15) and 9 + 15<br />

24 24<br />

d<br />

<br />

−7 − (−4) and −7 + 4<br />

−3<br />

−3<br />

<br />

a 21 − 13 21 + (−13) b −11 − 7 −11 + (−7) c 6 − (−13) 6 + 13 d −5 − (−1)<br />

−5 + 1.<br />

<br />

a 15 − 7 15 + (−7) b −14 − 8 −14 + (−8) c 4 − (−19) 4 + 19 d −4 − (−7) <br />

−4 + 7.


7−(−4−3) −9.<br />

<br />

7−(−4 − 3) −9<br />

<br />

7−(−7) −9<br />

14 − 9<br />

5<br />

8 − (−3 − 1) − 9.<br />

12 − (−9 − 6) − 14.<br />

<br />

<br />

<br />

<br />

a · b <br />

−3? 5, 3


5·3=15 −5(3) = −15<br />

5(−3) = −15 (−5)(−3) =15<br />

<br />

signs are the same, the product is positive.<br />

signs are different, the product is negative.<br />

15 ÷ 3 = 5 5 · 3 = 15. <br />

<br />

<br />

5·3=15 so 15÷3=5 −5(3) = −15 so −15 ÷ 3 = −5<br />

(−5)(−3) =15 so 15÷(−3) =−5 5(−3) = −15 so −15 ÷ (−3) =5


a −100 ÷ (−4) b 7·6 c 4(−8) d −27 ÷ 3.<br />

a<br />

<br />

−100 ÷ (−4)<br />

25<br />

b<br />

7·6<br />

42<br />

c<br />

4(−8)<br />

<br />

−32<br />

d<br />

−27÷3<br />

<br />

−9<br />

a −115 ÷ (−5) b 5 · 12 c 9(−7) d −63 ÷ 7.<br />

a −117 ÷ (−3) b 3 · 13 c 7(−4) d −42 ÷ 6.<br />

−1,<br />

<br />

<br />

−1<br />

−1a =−a<br />

−1


a (−2) 4 b −2 4 .<br />

<br />

−2 <br />

<br />

a<br />

(−2) 4<br />

<br />

<br />

<br />

(−2)(−2)(−2)(−2)<br />

4(−2)(−2)<br />

−8(−2)<br />

16<br />

b<br />

−2 4<br />

<br />

−(2·2·2·2)<br />

2 4 <br />

<br />

<br />

<br />

−(4·2·2)<br />

−(8·2)<br />

−16<br />

<br />

a (−3) 4 b −3 4 .<br />

<br />

a (−7) 2 b −7 2 .<br />

(−2) 4 −2 4 . <br />

<br />

<br />

a 8(−9) ÷ (−2) 3 b −30 ÷ 2 + (−3)(−7).<br />

a


8(−9) ÷ (−2) 3<br />

<br />

<br />

8(−9) ÷ (−8)<br />

−72 ÷ (−8)<br />

9<br />

b<br />

−30÷2+(−3)(−7)<br />

<br />

−15 + (−3)(−7)<br />

−15+21<br />

6<br />

<br />

a 12(−9) ÷ (−3) 3 b −27 ÷ 3 + (−5)(−6).<br />

<br />

a 18(−4) ÷ (−2) 3 b −32 ÷ 4 + (−2)(−7).<br />

<br />

<br />

<br />

<br />

4x 2 −2xy +3y 2 x =2,y =−1.<br />

<br />

<br />

<br />

<br />

<br />

<br />

3x 2 − 2xyx + 6y 2 x = 1, y = −2.<br />

<br />

4x 2 − xy<br />

+ 5y 2 x = −2, y = 3.


−12, 3.<br />

<br />

the sum of – 8 ––– and −12 increased by 3<br />

<br />

<br />

<br />

<br />

⎡<br />

⎣8+(−12) ⎤ ⎦ +3<br />

(−4) +3<br />

−1<br />

<br />

−16, <br />

<br />

−8 −12, <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

−9


15 <br />

<br />

<br />

−6 −15


, = <br />

<br />

a |−7|___ − |−7|<br />

b 6___ − |−6|<br />

c |−11|___ − 11<br />

d −(−13)___ − |−13|<br />

<br />

a −|−9|___|−9|<br />

b −8___|−8|<br />

c |−1|___ − 1<br />

d −(−14)___ − |−14|<br />

<br />

a −|2|___ − |−2|<br />

b −12___ − |−12|<br />

c |−3|___ − 3<br />

d |−19|___ − (−19)<br />

<br />

a −|−4|___ − |4|<br />

b 5___ − |−5|<br />

c −|−10|___ − 10<br />

d −|−0|___ − (−0)<br />

<br />

|15 − 7| − |14 − 6| |17 − 8| − |13 − 4| 18 − |2(8−3)|<br />

15 − |3(8−5)| 18 − |12 − 4(4−1) +3| 27 − |19 + 4(3−1) −7|<br />

10 − 3|9−3(3−1)|<br />

13 − 2|11 − 2(5−2)|<br />

<br />

<br />

<br />

a −7 + (−4)<br />

b −7+4<br />

c 7+(−4).<br />

<br />

a −5 + (−9)<br />

b −5+9<br />

c 5+(−9)<br />

48 + (−16)<br />

34 + (−19) −14 + (−12) +4 −17 + (−18) +6<br />

19 + 2(−3 + 8) 24 + 3(−5 + 9) <br />

a 13 − 7<br />

b −13 − (−7)<br />

c −13−7<br />

d 13 − (−7)


a 15 − 8<br />

b −15 − (−8)<br />

c −15−8<br />

d 15 − (−8)<br />

−17 − 42<br />

−58 − (−67)<br />

−14 − (−27) +9 64 + (−17) −9 a 44 − 28 b 44 + (−28)<br />

a 35 − 16 b 35 + (−16) a 27 − (−18) b 27 + 18 a 46 − (−37) b 46 + 37<br />

(2−7) − (3−8) (1−8) − (2−9) −(6−8) − (2−4)<br />

−(4−5) − (7−8) 25 − ⎡ ⎣10 − (3−12) ⎤ ⎦ 32 − ⎡ ⎣5−(15 − 20) ⎤ ⎦<br />

<br />

<br />

<br />

a −4 · 8<br />

b 13(−5)<br />

c −24 ÷ 6<br />

d −52 ÷ (−4)<br />

<br />

a −3 · 9<br />

b 9(−7)<br />

c 35 ÷ (−7)<br />

d −84 ÷ (−6)<br />

<br />

a −28 ÷ 7<br />

b −180 ÷ 15<br />

c 3(−13)<br />

d −1(−14)<br />

<br />

a −36 ÷ 4<br />

b −192 ÷ 12<br />

c 9(−7)<br />

d −1(−19)<br />

<br />

<br />

a (−2) 6 b −2 6 a (−3) 5 5<br />

b −3 5(−6) +7(−2) −3<br />

8(−4) +5(−4) −6 −3(−5)(6) −4(−6)(3)<br />

(8−11)(9−12) (6−11)(8−13) 26 − 3(2−7)<br />

23 − 2(4−6) 65 ÷ (−5) + (−28) ÷ (−7) 52 ÷ (−4) + (−32) ÷ (−8)<br />

9−2 ⎡ ⎣3−8(−2) ⎤ ⎦ 11 − 3 ⎡ ⎣7−4(−2) ⎤ ⎦ 8−|2−4(4−1) +3|<br />

7−|5−3(4−1) −6| 9−3|2(2−6) − (3−7)| 5−2|2(1−4) − (2−5)|<br />

(−3) 2 −24÷(8−2) (−4) 2 −32÷(12 − 4)


y + (−14) <br />

a y = −33 b y =30<br />

x + (−21) <br />

a x = −27 b x =44<br />

(x + y) 2 <br />

x =−3,y =14<br />

(y + z) 2 <br />

y =−3,z =15<br />

9a −2b −8 <br />

a =−6 b =−3<br />

7m −4n −2 <br />

m =−4 n =−9<br />

3x 2 −4xy +2y 2 <br />

x =−2,y =−3<br />

4x 2 − xy +3y 2 <br />

x =−3,y =−2<br />

<br />

<br />

−15, −8 −9,<br />

<br />

23<br />

<br />

a 10 −18<br />

b 11 −25<br />

<br />

a −5 −30<br />

b −6 −13<br />

−6 <br />

a b<br />

−13 <br />

c d<br />

<br />

<br />

<br />

<br />

<br />

−12°. <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

89°,<br />

<br />

−31°. <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

−$49 <br />

<br />

<br />

−8 <br />

−2 <br />

<br />

<br />

<br />

<br />

−4 3 = (−4) 3 ?


a <br />

b


2 3<br />

<br />

2 , <br />

3<br />

<br />

<br />

2<br />

3<br />

<br />

<br />

<br />

a , b ≠0 <br />

b<br />

<br />

b <br />

a <br />

<br />

<br />

<br />

b ≠0,c ≠0,<br />

a b = a · c<br />

b · c a · c<br />

b · c = a b .<br />

<br />

<br />

2 3.<br />

3<br />

10 15.<br />

15


− 315<br />

770 .<br />

<br />

<br />

− 69<br />

120 .<br />

<br />

− 120<br />

192 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

b ≠0, d ≠0, <br />

a · = ac<br />

b<br />

cd bd<br />

<br />

<br />

<br />

<br />

<br />

a 1 . 3=3 1 .


− 12 5 (−20x).<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

11 3 (−9a).<br />

<br />

13 7 (−14b).<br />

<br />

<br />

2 3 3 2 . 4 , <br />

1<br />

1 4 .<br />

<br />

<br />

b ≠0,c ≠0, d ≠0, <br />

a<br />

b ÷ d c = a b · dc<br />

<br />

b ≠0,<br />

c ≠0, d ≠0, <br />

<br />

−<br />

18 7 ÷ ⎛ ⎝ −14 ⎞<br />

27⎠ .


−<br />

27 7 ÷ ⎛ ⎝ − 35<br />

⎞<br />

36⎠<br />

.<br />

<br />

−<br />

14 5 ÷ ⎛ ⎝ − 15<br />

⎞<br />

28⎠<br />

.<br />

<br />

<br />

<br />

<br />

<br />

6<br />

7<br />

3<br />

<br />

3 4 ÷ 5 8 .<br />

3<br />

4<br />

5<br />

8<br />

x<br />

2<br />

5<br />

6<br />

3<br />

4<br />

5<br />

8<br />

<br />

x<br />

<br />

2<br />

xy.<br />

6


x<br />

2<br />

xy<br />

6<br />

<br />

<br />

<br />

<br />

x<br />

2 ÷ xy<br />

6<br />

x<br />

2 · 6xy<br />

x ·6<br />

2·xy<br />

x ·3·2<br />

2 · x · y<br />

<br />

3 y<br />

<br />

<br />

a<br />

8 .<br />

ab<br />

6<br />

<br />

<br />

p<br />

2<br />

pq .<br />

8<br />

<br />

<br />

<br />

<br />

c ≠0, <br />

a c + b c = a + b c<br />

and a c − b c = a − b c<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

7<br />

12<br />

+ 5 18 .


7<br />

12<br />

+ 11<br />

15 .<br />

<br />

13<br />

15 + 17<br />

20 .


a · = ac<br />

b<br />

cd bd<br />

<br />

<br />

<br />

a c + b c = a + b c<br />

<br />

<br />

<br />

a<br />

b ÷ d c = a b · dc<br />

<br />

<br />

<br />

a c − b c = a − b c<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a 5x<br />

6 − 3 10 b 5x<br />

6 · 310 .<br />

<br />

<br />

<br />

a<br />

<br />

<br />

<br />

5x<br />

6<br />

− 3 10<br />

<br />

6=2·3<br />

___________ 10=2·5<br />

LCD=2·3·5<br />

LCD = 30<br />

<br />

5x ·5<br />

6·5 − 3·3<br />

10 · 3<br />

25x<br />

30 − 9 30<br />

<br />

25x −9<br />

30<br />

<br />

b


5x<br />

6 · 310<br />

5x ·3<br />

6·10<br />

5 x ·3<br />

2·3·2·5<br />

x<br />

4<br />

5x<br />

6 − 10 3 , 5x<br />

6 · 310 .<br />

<br />

a 3a 4 − 8 9 b 3a 4 · 89 .<br />

<br />

a 4k<br />

5 − 1 6 b 4k<br />

5 · 16 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

−1<br />

3<br />

=− 1 3<br />

1<br />

−3 =−1 3<br />

negative<br />

positive = negative<br />

positive<br />

negative = negative<br />

<br />

−a<br />

b<br />

= a −b =−a b<br />

<br />

<br />

4(−3) +6(−2)<br />

.<br />

−3(2) −2


4(−3) +6(−2)<br />

−3(2) −2<br />

<br />

<br />

−12 + (−12)<br />

−6 − 2<br />

−24<br />

−8<br />

3<br />

<br />

<br />

8(−2) + 4(−3)<br />

.<br />

−5(2) + 3<br />

<br />

<br />

7(−1) + 9(−3)<br />

.<br />

−5(3) − 2<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎛<br />

⎝ 1 2<br />

⎞<br />

2⎠<br />

4+3 2.<br />

<br />

<br />

<br />

⎛1<br />

⎛ ⎞ 2<br />

⎝3<br />

⎠<br />

2 3 + 2 .


1 + 42<br />

⎛1<br />

⎛ ⎞ 2 .<br />

⎝4<br />

⎠<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

1<br />

<br />

2 + 2 3<br />

3<br />

4 − 1 .<br />

6<br />

<br />

<br />

⎛<br />

⎝ 1 2 + 2 ⎞<br />

3⎠<br />

⎛<br />

⎝ 3 4 − 1 ⎞<br />

6⎠<br />

<br />

<br />

<br />

<br />

<br />

⎛<br />

⎝ 3 6 + 4 ⎞<br />

6⎠<br />

⎛<br />

⎝ 9 12 − 2 ⎞<br />

12⎠<br />

⎛<br />

⎝ 7 ⎞<br />

6⎠<br />

⎛<br />

⎝ 7 ⎞<br />

12⎠<br />

7<br />

6 ÷ 7 12<br />

7<br />

6 ⋅ 12 7<br />

7 ⋅6⋅2<br />

6 ⋅7⋅1<br />

2<br />

<br />

1<br />

<br />

3 + 1 2<br />

3<br />

4 − 1 .<br />

3<br />

<br />

2<br />

<br />

3 − 1 2<br />

1<br />

4 + 1 .<br />

3


2x 2 y x = 1 4 y =−2 3 .<br />

<br />

<br />

<br />

<br />

2·2·4 <br />

<br />

<br />

<br />

3ab 2 a = − 2 3 b = − 1 2 .<br />

<br />

4c 3 d c = − 1 2 d = − 4 3 .


− 108<br />

63<br />

− 104<br />

48<br />

120<br />

252<br />

182<br />

294 14x 2<br />

21y<br />

<br />

24a<br />

32b 2<br />

− 210a2<br />

110b 2<br />

− 30x2<br />

105y 2<br />

<br />

<br />

− 3 ⎛<br />

4⎝ −4 ⎞<br />

9⎠<br />

− 3 8 · 415<br />

⎛ ⎝ −14 ⎞<br />

15⎠ ⎛ ⎝20<br />

9 ⎞<br />

⎠<br />

⎛ ⎝ − 10<br />

9 ⎞<br />

⎠ ⎛ ⎝ 25 ⎞<br />

33⎠<br />

⎛ ⎝ −63 ⎞ ⎛<br />

84⎠⎝ −44 ⎞<br />

90⎠<br />

⎛ ⎝ −33 ⎞<br />

60⎠ ⎛ ⎝ −40 ⎞<br />

88⎠<br />

3<br />

7 ·21n 5<br />

6 ·30m 3<br />

4<br />

÷ x<br />

11<br />

2<br />

5<br />

÷ y 9<br />

<br />

5 ÷ ⎛ 18 ⎝ −15 ⎞<br />

24⎠ 7 ÷ ⎛ 18 ⎝ −14 ⎞<br />

27⎠<br />

8u<br />

15 ÷ 12v<br />

25<br />

−15 ÷ ⎛ ⎝ −5 ⎞<br />

3⎠<br />

12r<br />

25 ÷ 18s<br />

35<br />

3<br />

4<br />

÷ (−12)<br />

<br />

<br />

− 8 21<br />

12<br />

35<br />

<br />

− 9 16<br />

33<br />

40<br />

<br />

− 4 5<br />

2<br />

<br />

5<br />

3<br />

10<br />

<br />

m<br />

3<br />

n<br />

2<br />

<br />

− 3 8<br />

− y<br />

12<br />

<br />

<br />

7<br />

12<br />

+ 5 8<br />

5<br />

12<br />

+ 3 8<br />

7<br />

12<br />

− 9 16<br />

7<br />

16<br />

− 5 12<br />

− 13<br />

30 + 25<br />

42<br />

− 23<br />

30 + 5 48


− 39<br />

56 − 22<br />

35<br />

− 3 4 − ⎛ ⎝ −4 ⎞<br />

5⎠<br />

− 33<br />

49 − 18<br />

35<br />

x<br />

3<br />

+ 1 4<br />

− 2 3 − ⎛ ⎝ −3 ⎞<br />

4⎠<br />

x<br />

5<br />

− 1 4<br />

<br />

a 2 3 + 1 6<br />

b 2 3 ÷ 1 6<br />

<br />

a − 2 5 − 1 8<br />

b − 2 5 · 18<br />

<br />

a 5n 6 ÷ 8 15<br />

b 5n 6 − 8 15<br />

<br />

a 3a 8 ÷ 7 12<br />

b 3a 8 − 7 12<br />

<br />

a − 5a 3 + ⎛ ⎝ −10 ⎞<br />

6 ⎠<br />

b − 5a 3 ÷ ⎛ ⎝ −10 ⎞<br />

6 ⎠<br />

<br />

a − 4x<br />

9 − 5 6<br />

b − 4k<br />

9 · 56<br />

<br />

a 2b 5 + 8 15<br />

b 2b 5 ÷ 8 15<br />

<br />

a − 3y<br />

8 − 4 3<br />

b − 3y<br />

8 · 43<br />

<br />

<br />

5·6−3·4<br />

4·5−2·3<br />

8·9−7·6<br />

5·6−9·2 5 2 −3 2<br />

3−5<br />

6 2 −4 2<br />

4−6<br />

<br />

7·4−2(8−5)<br />

9·3−3·5<br />

<br />

9·7−3(12 − 8)<br />

8·7−6·6<br />

<br />

9(8−2) −3(15 − 7)<br />

6(7−1) −3(17 − 9)<br />

<br />

8(9−2) −4(14 − 9)<br />

7(8−3) −3(16 − 9)<br />

2 3 +4 2<br />

2<br />

⎛<br />

⎝ 2 ⎞<br />

3⎠<br />

3 3 −3 2<br />

2<br />

⎛<br />

⎝ 3 ⎞<br />

4⎠<br />

<br />

⎛<br />

⎝ 3 2<br />

⎞<br />

5⎠<br />

⎛<br />

⎝ 3 2<br />

⎞<br />

7⎠<br />

<br />

⎛<br />

⎝ 3 2<br />

⎞<br />

4⎠<br />

⎛<br />

⎝ 5 2<br />

⎞<br />

8⎠<br />

2<br />

1<br />

3 + 1 5<br />

5<br />

1<br />

4 + 1 3<br />

<br />

7<br />

8 − 2 3<br />

1<br />

2 + 3 8<br />

<br />

3<br />

4 − 3 5<br />

1<br />

4 + 2 5<br />

<br />

<br />

− 3 8 ÷ ⎛ ⎝ − 10<br />

3 ⎞<br />

⎠ −<br />

12 3 ÷ ⎛ ⎝ −5 ⎞<br />

9⎠<br />

− 3 8 + 5 12


− 1 8 + 7 12<br />

− 7 15 − y 4<br />

− 3 8 − x<br />

11<br />

11<br />

12a · 9a<br />

16<br />

<br />

10y<br />

13 · 8<br />

15y<br />

1<br />

2<br />

+ 2 3 · 512<br />

1<br />

3<br />

+ 2 5 · 34<br />

1− 3 5 ÷ 1 10<br />

1− 5 6 ÷ 1 12<br />

3<br />

8<br />

− 1 6 + 3 4<br />

2<br />

5<br />

+ 5 8 − 3 4<br />

12 ⎛ ⎝20 9 − 15<br />

4 ⎞<br />

⎠<br />

8 ⎛ ⎝ 15<br />

16 − 5 ⎞<br />

6⎠<br />

<br />

5<br />

8 + 1 6<br />

19<br />

24<br />

<br />

1<br />

6 + 3 10<br />

14<br />

30<br />

⎛ ⎝ 5 9 + 1 ⎞<br />

6⎠ ÷ ⎛ ⎝ 2 3 − 1 ⎞<br />

2⎠ ⎛ ⎝ 3 4 + 1 ⎞<br />

6⎠ ÷ ⎛ ⎝ 5 8 − 1 ⎞<br />

3⎠<br />

<br />

<br />

<br />

7<br />

10<br />

− w <br />

<br />

5<br />

12<br />

− w <br />

2x 2 y 3 <br />

a w = 1 2 b w =−1 2<br />

8u 2 v 3 <br />

u =− 3 4 v =−1 2<br />

a w = 1 4 b w =−1 4<br />

a + b<br />

a − b <br />

a =−3,b =8<br />

x =− 2 3 y =−1 2<br />

r − s<br />

r + s <br />

r = 10, s =−5<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


0.1 =<br />

10<br />

1<br />

0.01 =<br />

100<br />

1<br />

0.001 = 1<br />

1000<br />

0.0001 = 1<br />

10,000<br />

is “one tenth”<br />

is “one hundredth”<br />

is “one thousandth”<br />

is “one ten-thousandth”


a b c <br />

<br />

18.379.<br />

a <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

b


c <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

6.582 a b c <br />

<br />

15.2175 a b c <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a −23.5 − 41.38 b 14.65 − 20.<br />

a


−23.5 − 41.38<br />

<br />

<br />

23.5<br />

______ +41.38<br />

<br />

5 = 50<br />

23.50<br />

10 100 0.5 = 0.50 ______ +41.38<br />

<br />

<br />

23.50<br />

______ +41.38<br />

64.88<br />

−23.5 − 41.38 = −64.88<br />

b<br />

<br />

14.65 − 20<br />

<br />

20<br />

______ −14.65<br />

<br />

<br />

<br />

20.00<br />

______ −14.65<br />

9 9<br />

1 10 10 10<br />

2 0 . 0 0<br />

______________<br />

−1 4 . 6 5<br />

5. 3 5<br />

14.65 − 20 = −5.35<br />

a −4.8 − 11.69 b 9.58 − 10.<br />

a −5.123 − 18.47 b 37.42 − 50.


(−3.9)(4.075).<br />

<br />

(−3.9)(4.075)<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

(−3.9)(4.075) = −15.8925<br />

<br />

−4.5(6.107).<br />

<br />

−10.79(8.12).


a b c <br />

<br />

<br />

<br />

a<br />

<br />

b<br />

<br />

c<br />

<br />

<br />

<br />

a b c <br />

<br />

a b c


−25.65 ÷ (−0.06).<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

−23.492 ÷ (−0.04).<br />

<br />

−4.11 ÷ (−0.12).<br />

<br />

<br />

<br />

0.03. <br />

<br />

0.03 = 3<br />

100


a 0.374 b − 5 8 <br />

a<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

b 5 8<br />

85. <br />

<br />

a 0.234 b − 7 8 <br />

<br />

a 0.024 b − 3 8


a <br />

b <br />

<br />

a<br />

<br />

b<br />

<br />

<br />

<br />

a <br />

b <br />

<br />

<br />

a <br />

b


n n 2 n <br />

8 2 <br />

11 2 <br />

<br />

n 2 = m, <br />

<br />

<br />

(−3) 2 =9 (−8) 2 =64 (−11) 2 = 121 (−15) 2 = 225<br />

10 2 = 100, <br />

<br />

<br />

n 2 = m, <br />

(−10) 2 = 100 −10 −10 <br />

m <br />

<br />

<br />

<br />

m m <br />

m = n 2 , m = n, n ≥0.<br />

<br />

m, <br />

<br />

100 = 10. <br />

− 100 = −10. − 100 <br />

<br />

a 25 b 121 c − 144.<br />

a<br />

<br />

5 2 =25 5<br />

25<br />

b


11 2 = 121 11<br />

121<br />

c<br />

− 144<br />

<br />

−12<br />

a 36 b 169 c − 225.<br />

a 16 b 196 c − 100.<br />

<br />

<br />

<br />

<br />

Counting numbers 1, 2, 3, 4, …..<br />

Whole numbers 0, 1, 2, 3, 4, ….<br />

Integers<br />

…. − 3, −2, −1, 0, 1, 2, 3, ….<br />

<br />

<br />

<br />

−1.2684 <br />

<br />

1 3 0. 3– . <br />

<br />

<br />

<br />

<br />

q p , q ≠0, <br />

<br />

<br />

q p , q ≠0.<br />

<br />

π <br />

<br />

<br />

π = 3.141592654...


−25? −25?<br />

() 2 = −25?<br />

−25. <br />

−25. <br />

<br />

<br />

,8, 5, 5.9, − 64, a b c d <br />

−7, 14 5<br />

e <br />

<br />

a 0, 1, 2, 3, …,


−7 −64=−8. <br />

−7, 8, −64.<br />

c −7, 8, −64 <br />

14 5 5.9 −7, 14 , 8, 5.9, −64.<br />

5<br />

d <br />

5 <br />

e <br />

<br />

−3, − 2, 0. 3 – , 9 5 , 4,<br />

49, a b c <br />

d e <br />

<br />

− 25, − 3 , −1, 6, 121, 2.041975..., a b c <br />

8<br />

d e <br />

<br />

<br />

1<br />

5 , − 4 5 ,3,7 4 , − 9 2 ,−5 8 3 <br />

−5 <br />

1 5 −4 5 . 1 5<br />

<br />

1 5 , 2 5 , 3 5 , 4 . <br />

5<br />

1<br />

5 .<br />

− 4 5 −1. −4 5 .<br />

7 4 , 9 2 , 8 . <br />

3<br />

<br />

7<br />

4 =13 −<br />

4<br />

9 2 =−41 2<br />

<br />

8<br />

3 =22 3<br />

<br />

<br />

4, 3 4 , − 1 4 ,−3,6 5 , − 5 2 , 7 3 .


4, −3.<br />

3 4 3 4<br />

<br />

3 4 . −1 4 .<br />

6 5 , − 5 2 ,<br />

7 . <br />

3<br />

6 5 =11 5 , − 5 2 =−21 2 , 7 3 =21 3 .<br />

<br />

−1, 1 3 , 6 5 , − 7 4 , 9 2 , 5, − 8 3 .<br />

<br />

−2, 2 3 , 7 5 , − 7 4 , 7 2 , 3, − 7 3 .<br />

<br />

<br />

<br />

a b −0.74.<br />

<br />

a <br />

10 4 , <br />

<br />

<br />

b −0.74 −<br />

100 74 , −1. <br />

−1.<br />

<br />

a 0.6 b −0.25.<br />

<br />

a 0.9 b −0.75.


a b c <br />

<br />

<br />

<br />

<br />

−16.53 − 24.38 −19.47 − 32.58 −38.69 + 31.47<br />

−29.83 + 19.76 72.5 − 100 86.2 − 100<br />

91.75 − (−10.462) 94.69 − (−12.678) 55.01 − 3.7<br />

59.08 − 4.6 2.51 − 7.4 3.84 − 6.1<br />

<br />

<br />

⎛ ⎝94.69)(−12.678) (−8.5)(1.69) (−5.18)(−65.23)<br />

(−9.16)(−68.34) (0.06)(21.75) (0.08)(52.45)<br />

(9.24)(10) (6.531)(10) (0.025)(100)<br />

(0.037)(100) (55.2)(1000) (99.4)(1000)<br />

<br />

$117.25 ÷ 48 $109.24 ÷ 36 1.44 ÷ (−0.3)<br />

−1.15 ÷ (−0.05) 5.2 ÷ 2.5 14 ÷ 0.35<br />

<br />

<br />

0.04 0.095<br />

−0.375<br />

<br />

17<br />

20<br />

− 18<br />

11<br />

17<br />

4<br />

− 310<br />

25


71% 150% 39.3%<br />

7.8%<br />

<br />

1.56 0.0625<br />

2.254<br />

<br />

<br />

64 169 144<br />

−4 − 100 − 121<br />

<br />

a b c d e <br />

<br />

<br />

<br />

−8, 0, 1.95286..., 12 5 , 36,9 −9, −3 4 —<br />

,− 9, 0.4 09 , 11,7<br />

9 6<br />

<br />

− 100, −7, − 8 3 , −1, 0.77, 31 4<br />

<br />

−6, − 5 ———<br />

, 0, 0.714285 ,2 1, 14<br />

2 5<br />

<br />

<br />

3<br />

10<br />

, 7 2 , 11 6 ,4 7<br />

10<br />

, 5 2 , 13 8 ,3 3<br />

4<br />

, − 3 4 ,12 3 ,−12 3 , 5 2 , − 5 2<br />

2<br />

5<br />

, − 2 5 ,13 4 ,−13 4 , 8 3 , − 8 3<br />

a 0.8 b −1.25 a −0.9 b −2.75<br />

a −1.6 b 3.25 a 3.1 b −3.65


a <br />

b


8+9 9+8, <br />

8+9=9+8. <br />

9·8 8·9 <br />

9·8=8·9. <br />

<br />

<br />

of Addition If a and b are real numbers, then a + b = b + a.<br />

of Multiplication If a and b are real numbers, then a · b = b · a.<br />

<br />

9−8 8−9 9−8≠8−9. <br />

<br />

12 ÷ 3 ≠ 3 ÷ 12, <br />

<br />

<br />

<br />

<br />

(7+8) +2=7+(8+2), <br />

<br />

⎛<br />

⎝ 5·13 ⎞<br />

⎠·3=5·⎛⎝ 1 3 ·3⎞ ⎠ , <br />

<br />

<br />

of Addition If a, b, and c are real numbers, then (a + b) + c = a + (b + c).<br />

of Multiplication If a, b, and c are real numbers, then (a · b) · c = a · (b · c).


(10 − 3) −2≠10−(3−2) (24 ÷ 4) ÷2≠24÷(4 ÷2)<br />

7−2≠10−1<br />

6÷2≠24÷2<br />

5≠9<br />

3≠12<br />

<br />

<br />

<br />

18p +6q +15p +5q.<br />

<br />

18p +6q +15p +5q<br />

<br />

<br />

18p +15p +6q +5q<br />

33p +11q<br />

<br />

23r + 14s + 9r +15s.<br />

37m + 21n + 4m − 15n.<br />

<br />

<br />

<br />

⎛ ⎝13 5 + 3 ⎞<br />

4⎠ + 1 4 .<br />

<br />

⎛<br />

⎝13 5 + 3 ⎞<br />

4⎠ + 1 4<br />

<br />

<br />

<br />

5<br />

13 + ⎛ ⎝ 3 4 + 1 ⎞<br />

4⎠<br />

5<br />

13 + ⎛ ⎝ 4 ⎞<br />

4⎠<br />

5<br />

13 +1<br />

1 5 13<br />

<br />

18<br />

13<br />

<br />

⎛ 7<br />

⎝ + 5 ⎞<br />

15 8⎠<br />

+ 3.<br />

8


⎛2 ⎝ + 7 ⎞<br />

9 12⎠<br />

+ 5 . 12<br />

<br />

<br />

a, a +0=a 0+a = a.<br />

<br />

a, a ·1=a <br />

1·a = a.<br />

<br />

<br />

of Addition For any real number a : a +0=a 0+a = a<br />

0 is the additive identity<br />

of Multiplication For any real number a : a ·1=a 1·a = a<br />

1 is the multiplicative identity<br />

<br />

<br />

−a a. <br />

<br />

a, a + (−a) =0.<br />

2 3 2<br />

3<br />

<br />

<br />

1 a <br />

a, a ≠0,a · 1a =1.<br />

<br />

<br />

of Addition For any real number a, a + (−a) =0<br />

−a is the additive inverse of a<br />

A number and its opposite add to zero.<br />

of Multiplication For any real number a, a ≠0, a · 1a =1.<br />

1<br />

a is the multiplicative inverse of a.<br />

A number and its reciprocal multiply to one.<br />

<br />

<br />

0÷3? <br />

<br />

0÷3=0.<br />

0÷3=0 0·3=0.<br />

0?


4? <br />

4÷0 <br />

<br />

<br />

<br />

<br />

a ·0=0 0·a = 0 The product of any number and 0 is 0.<br />

a ≠0<br />

a<br />

0<br />

0<br />

a = 0<br />

is undefined Division by zero is undefined.<br />

Zero divided by any real number, except itself, is zero.<br />

<br />

<br />

−84n + (−73n) +84n.<br />

<br />

−84n + (−73n) +84n<br />

<br />

<br />

<br />

<br />

−84n +84n + (−73n)<br />

0+(−73n)<br />

−73n<br />

−27a + (−48a) + 27a.<br />

<br />

39x + (−92x) + (−39x).<br />

<br />

<br />

<br />

7<br />

15 · 823 · 15 7 .


7<br />

15 · 823 · 15 7<br />

<br />

<br />

7<br />

15 · 15 7 · 823<br />

1· 823<br />

<br />

8<br />

23<br />

<br />

9<br />

16 · 549 · 16 9 .<br />

<br />

6<br />

17 · 11<br />

25 · 17 6 .<br />

<br />

<br />

<br />

a<br />

n +5 0<br />

10 − 3p<br />

, n ≠−5b , 10 − 3p ≠0.<br />

0<br />

<br />

a<br />

0<br />

n +5<br />

Zero divided by any real number except itself is 0. 0<br />

b<br />

Division by 0 is undefined.<br />

10 − 3p<br />

0<br />

undefined<br />

<br />

a<br />

m 0 , m ≠ −7 b 18 − 6c, 18 − 6c ≠ 0.<br />

+ 7 0<br />

<br />

a<br />

d − 0<br />

15 − 4q<br />

, d ≠ 4 b , 15 − 4q ≠ 0.<br />

4 0


If a, b, and c are real numbers, then<br />

a(b + c) = ab + ac<br />

(b + c)a = ba + ca<br />

a(b − c) = ab − ac<br />

(b − c)a = ba − ca<br />

<br />

<br />

3(x +4).<br />

<br />

3(x +4)<br />

3·x +3·4<br />

3x +12<br />

4(x + 2).<br />

6(x + 7).<br />

<br />

<br />

<br />

8 ⎛ ⎝ 3 8 x + 1 ⎞<br />

4⎠ .<br />

<br />

<br />

<br />

<br />

6 ⎛5 ⎝ y + 1 ⎞<br />

6 2⎞<br />

⎠ .<br />

<br />

12 ⎛1 ⎝ n + 3⎞ 3 4⎠ .


100 ⎛ ⎝0.3 + 0.25q ⎞ ⎠.<br />

<br />

<br />

<br />

100 ⎛ ⎝ ⎛ 0.7 + 0.15p ⎞ ⎠ ⎞ .<br />

<br />

100(0.04 + 0.35d).<br />

<br />

<br />

−11(4−3a).<br />

<br />

−11(4−3a)<br />

−11 · 4 − (−11) ·3a<br />

<br />

<br />

−44 − (−33a)<br />

−44+33a<br />

33a − 44. <br />

<br />

−5(2 − 3a).<br />

−7 ⎛ ⎝ ⎛ 8 − 15y ⎞ ⎠ ⎞ .<br />

<br />

<br />

− ⎛ ⎝y +5 ⎞ ⎠.


− ⎛ ⎝y +5 ⎞ ⎠<br />

−1 <br />

−1 ⎛ ⎝y +5 ⎞ ⎠<br />

−1 · y + (−1) ·5<br />

<br />

<br />

−y + (−5)<br />

−y −5<br />

−(z − 11).<br />

−(x − 4).<br />

<br />

<br />

<br />

<br />

8−2(x +3)<br />

<br />

<br />

<br />

8−2(x +3)<br />

8−2·x −2·3<br />

<br />

8−2x −6<br />

−2x +2<br />

9 − 3(x + 2).<br />

7x − 5(x + 4).<br />

<br />

4(x −8) − (x +3).


4(x −8) − (x +3)<br />

<br />

<br />

4x −32−x −3<br />

3x −35<br />

6(x − 9) − (x + 12).<br />

8(x − 1) − (x + 5).<br />

<br />

<br />

<br />

of addition If a, b are real numbers, then a + b = b + a<br />

of multiplication If a, b are real numbers, then a · b = b · a<br />

<br />

<br />

of addition If a, b, and c are real numbers, then (a + b) + c = a + (b + c)<br />

of multiplication If a, b, and c are real numbers, then (a · b) · c = a · (b · c)<br />

<br />

If a, b, and c are real numbers, then a(b + c) = ab + ac<br />

(b + c)a = ba + ca<br />

a(b − c) = ab − ac<br />

(b − c)a = ba − ca<br />

<br />

of addition For any real number a:<br />

a +0=a<br />

0 is the additive identity 0+a = a<br />

of multiplication For any real number a:<br />

a ·1=a<br />

1 is the multiplicative identity 1·a = a


of addition For any real number a, a + (−a) =0<br />

−a is the additive inverse of a<br />

A number and its opposite add to zero.<br />

of multiplication For any real number a, a ≠0 a · 1a =1<br />

1<br />

a is the multiplicative inverse of a<br />

A number and its reciprocal multiply to one.<br />

<br />

For any real number a, a ·0=0<br />

0·a =0<br />

For any real number a, a ≠0, 0 a =0<br />

For any real number a,<br />

a<br />

0<br />

is undefined


43m + (−12n) + (−16m) + (−9n)<br />

−22p +17q + ⎛ ⎝−35p ⎞ ⎠ + ⎛ ⎝−27q ⎞ ⎠<br />

3<br />

8<br />

g + 1 12 h + 7 8 g + 5 12 h<br />

5<br />

6<br />

a + 3 10 b + 1 6 a + 9 10 b<br />

6.8p + 9.14q + ⎛ ⎝−4.37p ⎞ ⎠ + ⎛ ⎝−0.88q ⎞ ⎠<br />

−24 · 7 · 38<br />

9.6m + 7.22n + (−2.19m) + (−0.65n)<br />

−36·11·49<br />

⎛ ⎝ 5 6 + 15<br />

8 ⎞<br />

⎠ + 15<br />

7<br />

⎛ ⎝ 11<br />

12 + 4 ⎞<br />

9⎠ + 5 9<br />

17(0.25)(4) 36(0.2)(5)<br />

⎡ ⎣2.48(12) ⎤ ⎦(0.5)<br />

⎡ ⎣9.731(4) ⎤ ⎦(0.75)<br />

12 ⎛ ⎝ 5 6 p⎞ ⎠<br />

20 ⎛ ⎝ 3 5 q⎞ ⎠<br />

<br />

<br />

19a +44−19a<br />

27c +16−27c<br />

1 + 7 2 8 + ⎛ ⎝ −1 ⎞<br />

2⎠ 2 + 5 5 12 + ⎛ ⎝ −2 ⎞<br />

5⎠<br />

10(0.1d)<br />

3<br />

20 · 49<br />

11 · 20 3<br />

100 ⎛ ⎝0.01p ⎞ ⎠<br />

13<br />

18 · 25 7 · 18<br />

13<br />

0<br />

u − 4.99 , u ≠ 4.99 0÷⎛ ⎝ y − 1 ⎞<br />

6⎠ , x ≠ 1 6<br />

32 − 5a<br />

0<br />

, 32 − 5a ≠0 28 − 9b, 28 − 9b ≠0<br />

0<br />

⎛ ⎝ 3 4 + 10 9 m⎞ ⎠ ÷0, 3 4 + 10 9 m ≠0 ⎛ ⎝16 5 n − 3 ⎞<br />

7⎠ ÷0, 5<br />

16 n − 3 7 ≠0<br />

<br />

<br />

8 ⎛ ⎝4y +9 ⎞ ⎠ 9(3w +7) 6(c −13)


7 ⎛ ⎝y −13 ⎞ ⎠<br />

1<br />

4<br />

⎛<br />

⎝3q +12 ⎞ ⎠ 1<br />

5<br />

(4m +20)<br />

9 ⎛ ⎝ 5 9 y − 1 ⎞<br />

3⎠ 10 ⎛ ⎝10 3 x − 2 ⎞<br />

5⎠<br />

12 ⎛ ⎝ 1 4 + 2 3 r⎞ ⎠<br />

12 ⎛ ⎝ 1 6 + 3 4 s⎞ ⎠<br />

15 · (4d +10) 18 · (15h +24)<br />

35<br />

56<br />

r(s −18) u(v −10) ⎛ ⎝y +4 ⎞ ⎠p<br />

(a +7)x −7 ⎛ ⎝4p +1 ⎞ ⎠ −9(9a +4)<br />

−3(x −6) −4 ⎛ ⎝q −7 ⎞ ⎠ −(3x −7)<br />

− ⎛ ⎝5p −4 ⎞ ⎠ 16 − 3 ⎛ ⎝y +8 ⎞ ⎠ 18 − 4(x +2)<br />

4−11(3c −2) 9−6(7n −5) 22 − (a +3)<br />

8−(r −7) (5m −3) − (m +7) ⎛ ⎝4y −1 ⎞ ⎠ − ⎛ ⎝y −2 ⎞ ⎠<br />

9(8x −3) − (−2) 4(6x −1) − (−8) 5(2n +9) +12(n −3)<br />

9(5u +8) +2(u −6) 14(c −1) −8(c −6) 11(n −7) −5(n −1)<br />

6 ⎛ ⎝7y +8 ⎞ ⎠ − ⎛ ⎝30y −15 ⎞ ⎠ 7(3n +9) − (4n −13)<br />

<br />

<br />

<br />

8 ⎛ ⎝ x − 1 ⎞<br />

4⎠ <br />

<br />

<br />

<br />

4($5.97) <br />

$5.97 6 − 0.03<br />

<br />

<br />

a <br />

b


0 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a ,<br />

b<br />

b ≠0, <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a · b = m, <br />

<br />

a , b ≠0, <br />

b<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

0 <br />

<br />

a , b ≠0, <br />

b<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

q p , q ≠0.


n 2 = m, <br />

<br />

n 2 = m, <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a = b


a ≠ b<br />

a < b<br />

a ≤ b<br />

a > b<br />

a ≥ b<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

Parentheses ()<br />

Brackets []<br />

Braces {}<br />

<br />

a n <br />

a n n th


a <br />

<br />

<br />

<br />

<br />

b<br />

<br />

<br />

<br />

<br />

<br />

a b<br />

<br />

<br />

<br />

<br />

<br />

a + b<br />

a − b<br />

a · b, ab, a(b), (a)(b)<br />

2a<br />

a ÷ b, a/b, a b , ba<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

−a means the opposite of the number a<br />

The notation −a is read as “the opposite of a.”<br />

<br />

<br />

|n| |n| ≥0 <br />

<br />

<br />

<br />

a − b = a + (−b)<br />

Parentheses () Braces {}<br />

Brackets [] Absolute value ||


−1<br />

−1a =−a<br />

−1 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

b ≠0,c ≠0, <br />

a<br />

b = a b · · c<br />

c and a b · · c<br />

c = a b .<br />

<br />

<br />

<br />

<br />

<br />

<br />

b ≠0, d ≠0, <br />

a · = ac<br />

b<br />

cd bd .<br />

<br />

<br />

b ≠0,c ≠0, d ≠0, <br />

a<br />

b ÷ c d = a b · dc .<br />

<br />

<br />

c ≠0, <br />

a c + b c = a + c<br />

b and a c − b c = a − c b .


−a<br />

b<br />

= a −b =−a b .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

5?


m <br />

m = n 2 , m = n, n ≥0.<br />

m, <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

of addition If a, b are real numbers, then a + b = b + a<br />

of multiplication If a, b are real numbers, then a · b = b · a<br />

<br />

<br />

of addition If a, b, and c are real numbers, then (a + b) + c = a + (b + c)<br />

of multiplication If a, b, and c are real numbers, then (a · b) · c = a · (b · c)


If a, b, and c are real numbers, then a(b + c) = ab + ac<br />

(b + c)a = ba + ca<br />

a(b − c) = ab − ac<br />

(b − c)a = ba − ca<br />

<br />

of addition For any real number a:<br />

a +0=a<br />

0 is the additive identity 0+a = a<br />

of multiplication For any real number a:<br />

a ·1=a<br />

1 is the multiplicative identity 1·a = a<br />

<br />

of addition For any real number a, a + (−a) =0<br />

−a is the additive inverse of a<br />

A number and its opposite add to zero.<br />

of multiplication For any real number a, a ≠0 a · 1a =1<br />

1<br />

a is the multiplicative inverse of a<br />

A number and its reciprocal multiply to one.<br />

<br />

For any real number a, a ·0=0<br />

0·a =0<br />

For any real number a, a ≠0, 0 a =0<br />

For any real number a,<br />

a<br />

0<br />

is undefined<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

24 ÷ 3 + 4(5−2) 7+3 ⎡ ⎣6−4(5−4) ⎤ ⎦ −3 2


x =4, a x 3 b 5 x c 2x 2 −5x +3 2x 2 −4xy −3y 2 x =3, y =1<br />

<br />

<br />

12y +7+2y −5 14x 2 −9x +11−8x 2 +8x −6<br />

<br />

<br />

<br />

a 4ab 2 7a 3 b 2<br />

b 6y 2 3y<br />

c 5x<br />

d 5y 8y 2<br />

<br />

a y <br />

b y <br />

<br />

<br />

n <br />

<br />

<br />

<br />

<br />

, = <br />

<br />

a −|7|___ − |−7|<br />

b −8___ − |−8|<br />

c |−13|___ − 13<br />

d |−12|___ − (−12)<br />

<br />

9−|3(4−8)| 12 − 3|1−4(4−2)|<br />

<br />

<br />

−12 + (−8) +7 <br />

a 15 − 7<br />

b −15 − (−7)<br />

c −15−7<br />

d 15 − (−7)<br />

−11 − (−12) +5<br />

a 23 − (−17) b 23 + 17 −(7−11) − (3−5)


a −27÷9 b 120 ÷ (−8)<br />

c 4(−14) d −1(−17)<br />

<br />

<br />

a (−7) 3 3<br />

b −7 (7−11)(6−13) 63 ÷ (−9) + (−36) ÷ (−4)<br />

6−3|4(1−2) − (7−5)| (−2) 4 −24÷(13 − 5)<br />

<br />

(y + z) 2 <br />

y =−4,z =7<br />

3x 2 −2xy +4y 2 <br />

x =−2,y =−3<br />

<br />

<br />

−4 −9, a <br />

<br />

−8 b −25<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

204<br />

228 − 270x3<br />

198y 2<br />

<br />

<br />

⎛<br />

⎝ −14 ⎞<br />

15⎠ ⎛ ⎝ 10 ⎞<br />

21⎠ 6x<br />

25 ÷ 9y<br />

20 <br />

− 4 9<br />

8<br />

21<br />

<br />

<br />

5 + 7 11<br />

18 12<br />

36 − 15<br />

48<br />

a 5 8 + 3 4 b 5 8 ÷ 3 4


a − 3y<br />

10 − 5 6 b − 3y<br />

10 · 56<br />

<br />

<br />

<br />

4·3−2·5<br />

−6·3+2·3<br />

<br />

4(7−3) −2(4−9)<br />

−3(4+2) +7(3−6)<br />

4 3 −4 2<br />

2<br />

⎛<br />

⎝ 4 ⎞<br />

5⎠<br />

<br />

<br />

4x 2 y 2 <br />

a + b<br />

x = 2 a − b <br />

3 y =−3 4<br />

a =−4, b =6<br />

<br />

<br />

6.738 <br />

a b c <br />

<br />

<br />

<br />

−23.67 + 29.84 54.3 − 100 79.38 − (−17.598)<br />

<br />

<br />

(−2.8)(3.97) (−8.43)(−57.91) (53.48)(10)<br />

(0.563)(100) $118.35 ÷ 2.6 1.84 ÷ (−0.8)<br />

<br />

<br />

0.65 −9.6<br />

<br />

− 5 14<br />

8<br />

11<br />

<br />

2.43 0.0475<br />

<br />

<br />

289 −121<br />

<br />

a b c d e


−8, 0, 1.95286..., 12 5 , 36,9<br />

<br />

<br />

3<br />

4<br />

, − 3 4 ,11 3 ,−12 3 , 7 2 , − 5 2<br />

a 3.2 b −1.35<br />

<br />

<br />

<br />

5x + 5 8 12 y + 1 8 x + 12 7 y −32 · 9 · 58<br />

<br />

⎛<br />

⎝ 11<br />

15 + 3 ⎞<br />

8⎠ + 5 8<br />

<br />

<br />

4 + 8 7 15 + ⎛ ⎝ −4 ⎞<br />

7⎠<br />

13 · · 15<br />

15<br />

917 13<br />

0<br />

x −3 , x ≠3<br />

<br />

5x −7,5x −7≠0<br />

0<br />

<br />

<br />

8(a −4)<br />

12 ⎛ ⎝ 2 3 b + 5 ⎞<br />

6⎠<br />

18 · (2x −5) 56<br />

(x −5)p −4 ⎛ ⎝y −3 ⎞ ⎠ 12 − 6(x +3)<br />

6(3x −4) − (−5) 5 ⎛ ⎝2y +3 ⎞ ⎠ − ⎛ ⎝4y −1 ⎞ ⎠


756.<br />

<br />

5n +8+2n −1<br />

x =−2 <br />

y =3: |3x −4y|<br />

6<br />

<br />

<br />

a <br />

b −8 −3 <br />

<br />

<br />

<br />

<br />

n<br />

<br />

<br />

<br />

28.1458 <br />

<br />

a b <br />

<br />

a<br />

11 5 b 1.15 <br />

<br />

<br />

3 5 , 2.8, and − 5 2 <br />

<br />

<br />

8+3 ⎡ ⎣6−3(5−2) ⎤ ⎦ −4 2 −(4−9) − (9−5) 56 ÷ (−8) + (−27) ÷ (−3)<br />

16 − 2|3(1−4) − (8−5)| −5 + 2(−3) 2 −9 180<br />

204<br />

− 7 18 + 5 12<br />

<br />

4 ÷ ⎛ 5 ⎝ −12 ⎞<br />

25⎠<br />

9−3·9<br />

15 − 9<br />

<br />

4 ⎛ ⎝−3+2(3−6) ⎞ ⎠<br />

3 ⎛ ⎝11 − 3(2+3) ⎞ ⎠<br />

5<br />

13 ·47·13 5 <br />

− 5 9<br />

10<br />

21<br />

−4.8 + (−6.7) 34.6 − 100 −12.04 · (4.2)<br />

−8 ÷ 0.05 −121 <br />

⎛<br />

⎝13 8 + 5 ⎞<br />

7⎠ + 2 7<br />

5x + ⎛ ⎝−8y ⎞ ⎠ −6x +3y a 0 9 b 11 0<br />

−3(8x −5)<br />

<br />

6 ⎛ ⎝3y −1 ⎞ ⎠ − ⎛ ⎝5y −3 ⎞ ⎠


3 (12x + 20).<br />

2


5 − 2(n + 1).<br />

<br />

<br />

5 6 1 4 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

5y +3=10y −4.<br />

a y = 3 5 b y = 7 5<br />

<br />

<br />

<br />

a<br />

<br />

<br />

y = 3 5 y = 3 5<br />

5y +3=10y −4.<br />

b


y = 7 5 y = 7 5<br />

5y +3=10y −4.<br />

9y + 2 = 6y + 3.<br />

a y = 4 3 b y = 1 3<br />

4x − 2 = 2x + 1.<br />

a x = 3 2 b x = − 1 2<br />

<br />

<br />

a ≠0, <br />

ax + b =0<br />

<br />

<br />

<br />

<br />

<br />

7(n −3) − 8 = −15


2(m − 4) + 3 = −1.<br />

5(a − 3) + 5 = −10.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

2 (3m −6)=5−m.<br />

3


m =3.<br />

<br />

1 (6u + 3) = 7 − u.<br />

3<br />

<br />

2 (9x − 12) = 8 + 2x.<br />

3<br />

<br />

<br />

<br />

<br />

4(x −1) −2=5(2x +3) +6.<br />

<br />

<br />

<br />

4x <br />

10 > 4.<br />

<br />

<br />

<br />

<br />

<br />

<br />

x =− 9 2 .


6 ⎛<br />

⎝p<br />

− 3 ⎞ ⎠ − 7 = 5 ⎛ ⎝4p + 3 ⎞ ⎠ − 12.<br />

8 ⎛ ⎝ ⎛ q + 1 ⎞ ⎠ ⎞ − 5 = 3 ⎛ ⎝ ⎛ 2q − 4 ⎞ ⎠ ⎞ − 1.<br />

<br />

10 ⎡ ⎣3−8(2s −5) ⎤ ⎦ =15(40 − 5s).<br />

<br />

<br />

<br />

<br />

160s <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

s =−2.<br />

6 ⎡ ⎣ ⎡ 4 − 2 ⎛ ⎝ ⎛ 7y − 1 ⎞ ⎠ ⎞⎤ ⎦⎤ = 8 ⎛ ⎝ ⎛ 13 − 8y ⎞ ⎠ ⎞ .<br />

<br />

12 ⎡ ⎣ ⎡ 1 − 5(4z − 1) ⎤ ⎦ ⎤ = 3(24 + 11z).<br />

<br />

7x + 8 = −13 <br />

−3,


7y +14=7 ⎛ ⎝y +2 ⎞ ⎠. <br />

<br />

<br />

<br />

7y y’s <br />

<br />

14 = 14 <br />

7y +14=7 ⎛ ⎝y +2 ⎞ ⎠ <br />

<br />

<br />

<br />

<br />

−8z =−8z +9?<br />

<br />

8z <br />

z’s <br />

0≠9.<br />

−8z =−8z +9 0=9. −8z =−8z +9 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

6(2n −1) +3=2n −8+5(2n +1).


12n <br />

<br />

<br />

<br />

<br />

<br />

<br />

4 + 9(3x − 7) = −42x − 13 + 23(3x − 2).<br />

<br />

<br />

8(1 − 3x) + 15(2x + 7) = 2(x + 50) + 4(x + 3) + 1.<br />

<br />

<br />

8+3(a −4)=0.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a = 4 3 .<br />

<br />

a = 4 3 .<br />

<br />

<br />

11 ⎛ ⎝ ⎛ q + 3 ⎞ ⎠ ⎞ − 5 = 19.


6+14(k −8) = 95.<br />

<br />

<br />

5m +3(9+3m) =2(7m −11).<br />

<br />

<br />

<br />

14m <br />

<br />

27 ≠ −22.<br />

<br />

<br />

<br />

<br />

12c + 5(5 + 3c) = 3(9c − 4).<br />

<br />

<br />

4(7d + 18) = 13(3d − 2) − 11d.


1<br />

12<br />

x + 5 6 = 3 4 .<br />

<br />

<br />

1 4 x + 1 2 = 5 8 .<br />

<br />

1 8 x + 1 2 = 1 4 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

5= 1 2 y + 2 3 y − 3 4 y.


y = 12.<br />

<br />

7 = 1 2 x + 3 4 x − 2 3 x.<br />

<br />

−1 = 1 2 u + 1 4 u − 2 3 u.<br />

<br />

<br />

1 2 ⎛ ⎝y −5 ⎞ ⎠ = 1 4 ⎛ ⎝y −1 ⎞ ⎠.


y =9.<br />

<br />

<br />

1 5 (n + 3) = 1 (n + 2).<br />

4<br />

<br />

1 2 (m − 3) = 1 (m − 7).<br />

4<br />

<br />

<br />

<br />

4q +3<br />

+6=<br />

2<br />

<br />

3q +5<br />

4


q =−5.<br />

<br />

<br />

3r + 5<br />

6<br />

+1= 4r 3<br />

+ 3 .<br />

<br />

2s + 3<br />

2<br />

+ 1 = 3s 4<br />

+ 2 .<br />

<br />

0.7 =<br />

10 7 0.29 = 29 . <br />

100<br />

<br />

<br />

<br />

<br />

<br />

0.25x + 0.05(x + 3) = 2.85.<br />

<br />

<br />

0.25 =<br />

100 25 , 0.05 = 100 5 , 2.85 = 2 85<br />

100 .


x =9 <br />

0.25n + 0.05(n + 5) = 2.95.<br />

0.10d + 0.05(d − 5) = 2.15.


6y +10=12y<br />

4x +9=8x<br />

a y = 5 3 b y =−1 2<br />

a x =− 7 8 b x = 9 4<br />

8u −1=6u<br />

a u =− 1 2 b u = 1 2<br />

9v −2=3v<br />

a v =− 1 3 b v = 1 3<br />

<br />

15 ⎛ ⎝y −9 ⎞ ⎠ = −60 −16(3n +4) =32<br />

−(w −12) =30 −(t −19) =28<br />

51 + 5(4 − q) =56 −6+6(5−k) =15<br />

3(10 − 2x)+54=0<br />

−2(11 − 7x)+54=4<br />

2<br />

3<br />

(9c −3) =22 3<br />

5<br />

(10x −5) =27<br />

1<br />

5<br />

(15c + 10) = c +7 1<br />

4<br />

(20d + 12) = d +7<br />

3(4n −1)−2=8n +3 9(2m −3)−8=4m +7<br />

12+2(5−3y) =−9 ⎛ ⎝y −1 ⎞ ⎠ −2 −15 + 4(2 − 5y) =−7 ⎛ ⎝y −4 ⎞ ⎠ +4<br />

5 + 6(3s −5)=−3+2(8s −1) −12+8(x −5)=−4+3(5x −2)<br />

4 ⎛ ⎝p −4 ⎞ ⎠ − ⎛ ⎝p +7 ⎞ ⎠ =5 ⎛ ⎝p −3 ⎞ ⎠ 3(a −2) − (a +6) =4(a −1)<br />

4 ⎡ ⎣5−8(4c −3) ⎤ ⎦ =12(1−13c) −8 5 ⎡ ⎣9−2(6d −1) ⎤ ⎦ =11(4−10d) − 139<br />

3 ⎡ ⎣−9 + 8(4h −3) ⎤ ⎦ =2(5−12h) −19<br />

3 ⎡ ⎣−14 + 2(15k −6) ⎤ ⎦ =8(3−5k) −24<br />

<br />

5 ⎡ ⎣2(m +4) +8(m −7) ⎤ ⎦ =2 ⎡ ⎣3(5+m) − (21 − 3m) ⎤ ⎦<br />

<br />

10 ⎡ ⎣5(n +1) +4(n −1) ⎤ ⎦ =11 ⎡ ⎣7(5+n) − (25 − 3n) ⎤ ⎦<br />

<br />

<br />

<br />

23z +19=3(5z −9) +8z +46 15y +32=2 ⎛ ⎝10y −7 ⎞ ⎠ −5y +46


18(5j − 1) + 29 = 47 24(3d − 4) + 100 = 52<br />

22(3m −4)=8(2m +9) 30(2n −1)=5(10n +8)<br />

7v +42=11(3v +8) −2(13v −1) 18u −51=9(4u +5) −6(3u −10)<br />

45(3y −2)=9 ⎛ ⎝15y −6 ⎞ ⎠ 60(2x −1)=15(8x +5)<br />

9(14d +9)+4d =13(10d +6) +3 11(8c +5)−8c =2(40c +25) +5<br />

<br />

<br />

1<br />

4<br />

x − 1 2 =−3 4<br />

3<br />

4<br />

x − 1 2 = 1 4<br />

5<br />

6<br />

y − 2 3 =−3 2<br />

5<br />

6<br />

y − 1 3 =−7 6<br />

1<br />

2<br />

a + 3 8 = 3 4<br />

5<br />

8<br />

b + 1 2 =−3 4<br />

2= 1 3 x − 1 2 x + 2 3 x<br />

2=3 5 x − 1 3 x + 2 5 x<br />

1<br />

3<br />

w + 5 4 = w − 1 4<br />

1<br />

2<br />

a − 1 4 = 1 6 a + 1 12<br />

1<br />

3<br />

b + 1 5 = 2 5 b − 3 5<br />

1<br />

3<br />

x + 2 5 = 1 5 x − 2 5<br />

1<br />

4<br />

⎛<br />

⎝p −7 ⎞ ⎠ = 1 3 ⎛ ⎝p +5 ⎞ ⎠<br />

1<br />

5<br />

⎛<br />

⎝q +3 ⎞ ⎠ = 1 2 ⎛ ⎝q −3 ⎞ ⎠<br />

1<br />

2<br />

(x +4) = 3 4<br />

1<br />

3<br />

(x +5) = 5 6<br />

4n +8<br />

4<br />

3x +4<br />

2<br />

= n 3<br />

+1=<br />

5x +10<br />

8<br />

<br />

<br />

3p +6<br />

3<br />

10y −2<br />

3<br />

= p 2<br />

+3=<br />

10y +1<br />

9<br />

7u −1<br />

4<br />

−1=<br />

4u +8<br />

5<br />

3v −6<br />

2<br />

+5=<br />

11v −4<br />

5<br />

<br />

0.4x + 0.6 = 0.5x − 1.2 0.7x + 0.4 = 0.6x + 2.4<br />

0.9x − 1.25 = 0.75x + 1.75 1.2x − 0.91 = 0.8x + 2.29<br />

0.05n + 0.10(n +8) = 2.15 0.05n + 0.10(n +7) = 3.55<br />

0.10d + 0.25(d +5) = 4.05 0.10d + 0.25(d +7) = 5.25


2L +2(L − 2.5) = 74.<br />

<br />

<br />

<br />

<br />

0.49s + 0.21(s − 8) = 22.82 <br />

<br />

<br />

<br />

<br />

<br />

3−7(y −4)=38? <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

0.35x + 2.1 = 3.85, <br />

<br />

<br />

a <br />

b


s =


7·4+8 = ? 36<br />

28 + 8 = ? 36<br />

36 = 36 ✓<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

n =1 st <br />

n −9=2 nd


−12 −3?<br />

−3−9 = ? −12<br />

−12 = −12 ✓<br />

−15?<br />

−3 + (−12) = ? −15<br />

−15 = −15 ✓<br />

−3 −12.<br />

<br />

<br />

<br />

<br />

<br />

<br />

1, 2, 3, 4<br />

−10, −9, −8, −7<br />

150, 151, 152, 153<br />

<br />

n +1. n +1, n +1+1, n +2.<br />

n<br />

1 st integer<br />

n +1<br />

2 nd consecutive integer<br />

n +2 3 rd consecutive integer etc.<br />

<br />

<br />

−54.


n =1 st <br />

n +1=2 nd <br />

n +2=3 rd <br />

−54.<br />

<br />

<br />

<br />

<br />

<br />

−19 + ⎛ ⎝ −18 ⎞ ⎠ + ⎛ ⎝ −17 ⎞ ⎠ = −54<br />

−54 = −54 ✓<br />

<br />

<br />

−17, −18, −19.<br />

−96.<br />

−36.<br />

<br />

<br />

<br />

24, 26, 28<br />

−12, −10, −8<br />

n +2. <br />

n +2+2 n +4.<br />

n<br />

1 st even integer<br />

n +2<br />

2 nd consecutive even integer<br />

n +4 3 rd consecutive even integer etc.<br />

<br />

<br />

63, 65, 67<br />

n, n +2,n +4


n<br />

1 st odd integer<br />

n +2<br />

2 nd consecutive odd integer<br />

n +4 3 rd consecutive odd integer etc.<br />

<br />

<br />

<br />

<br />

<br />

120 <br />

<br />

<br />

<br />

<br />

<br />

Let n =1 st even integer.<br />

n +2= 2 nd consecutive even integer<br />

n +4= 3 rd consecutive even integer<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

The sum of the three even integers is 120.<br />

n + n +2+n + 4 = 120<br />

n + n +2+n + 4 = 120<br />

3n + 6 = 120<br />

3n = 114<br />

n =38<br />

1 st integer<br />

n +2<br />

38 + 2<br />

40<br />

n +4<br />

38 + 4<br />

42<br />

2 nd integer<br />

3 rd integer<br />

<br />

38 + 40 + 42 = ? 120<br />

120 = 120 ✓<br />

<br />

<br />

<br />

<br />

−24.


h +2h − 16,000 = 110,000<br />

h +2h − 16,000 = 110,000<br />

3h − 16,000 = 110,000<br />

3h = 126,000<br />

h = 42,000<br />

<br />

2h − 16,000 amount wife earns<br />

2(42,000) − 16,000<br />

84,000 − 16,000<br />

68,000<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a b c


a<br />

<br />

<br />

<br />

<br />

b<br />

<br />

<br />

<br />

<br />

c<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a b c <br />

<br />

a b a


a = <br />

<br />

<br />

<br />

<br />

<br />

1 4 <br />

1 4 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

p = <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

25% <br />

25% <br />

<br />

25%


change = new amount − original amount<br />

<br />

change is what percent of the original amount?<br />

<br />

<br />

<br />

<br />

46 − 36 = 10<br />

<br />

<br />

p = <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

27.8%


amount of discount = discount rate · original price<br />

sale price = original amount – discount price<br />

<br />

<br />

amount of mark-up = mark-up rate · original price<br />

list price = original cost + mark-up<br />

<br />

<br />

a <br />

b <br />

a<br />

<br />

<br />

<br />

<br />

m = <br />

<br />

<br />

<br />

<br />

<br />

<br />

b<br />

<br />

<br />

<br />

p =


a b <br />

<br />

<br />

a b <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

I = interest<br />

I = Prt where P = principal<br />

r = rate<br />

t = time<br />

<br />

I = Prt. <br />

<br />

<br />

<br />

<br />

<br />

I = ?<br />

P = $950<br />

r = 3%<br />

t = 5 years


Let I = interest.<br />

I = Prt<br />

I = (950)(0.03)(5)<br />

I = 142.5<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

6.5%.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

I = $1500<br />

P = $7500<br />

r = ?<br />

t = 5 years


Let r = rate of interest.<br />

I = Prt<br />

1,500 = (7,500)r(5)<br />

1,500 = 37,500r<br />

0.04 = r<br />

4% = r<br />

<br />

I = Prt<br />

1,500 = ? (7,500)(0.04)(5)<br />

1,500 = 1,500 ✓<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

I = 4,866.25<br />

P = ?<br />

r = 8.5%<br />

t = 5 years


Let P = principal borrowed.<br />

I = Prt<br />

4,866.25 = P(0.085)(5)<br />

4,866.25 = 0.425P<br />

11,450 = P<br />

<br />

I = Prt<br />

4,866.25 = ? (11,450)(0.085)(5)<br />

4,866.25 = 4,866.25 ✓


−36.<br />

<br />

−3.<br />

<br />

<br />

<br />

<br />

<br />

−213.<br />

<br />

<br />

<br />

<br />

<br />

<br />

−267.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b <br />

a <br />

a <br />

b <br />

a <br />

a <br />

b <br />

a <br />

<br />

a <br />

b <br />

a


a b <br />

<br />

<br />

<br />

<br />

a b <br />

<br />

<br />

<br />

<br />

<br />

a b


a <br />

b


2(x + 3) x = 5.<br />

<br />

<br />

<br />

<br />

<br />

<br />

1 bh b = 14 h = 9.<br />

2<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

V = 1 3 πr2 h <br />

<br />

<br />

<br />

V = 1 3 πr2 h <br />

<br />

<br />

<br />

<br />

πr 2 .


h = 3V<br />

πr 2.<br />

<br />

A = 1 bh <br />

2<br />

<br />

A = 1 bh <br />

2<br />

<br />

<br />

<br />

C = 5 (F − 32) <br />

9<br />

<br />

<br />

<br />

<br />

<br />

F = 9 C +32 <br />

5<br />

<br />

F = 9 C + 32 <br />

5<br />

<br />

A = 1 h(b + B) <br />

2<br />

<br />

<br />

S =2πr 2 +2πrh <br />

<br />

<br />

2πr 2 <br />

<br />

2πr.


A = P + Prt <br />

<br />

A = P + Prt <br />

<br />

<br />

<br />

8x +7y =15 <br />

<br />

<br />

6x <br />

<br />

<br />

<br />

<br />

4x + 7y = 9 <br />

<br />

5x + 8y = 1 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

A = 1 bh, <br />

2


h = <br />

<br />

<br />

<br />

A = 1 2 bh<br />

<br />

<br />

<br />

126 = 1 2 ·18·h<br />

126 = 9h<br />

14 = h<br />

<br />

A = 1 2 bh<br />

126 = ? 1 ·18·14 2<br />

126 = 126 ✓


180°. <br />

m ∠ A + m ∠ B + m ∠ C = 180. <br />

90° <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

Let a = 1 st angle.<br />

a +40 = 2 nd angle<br />

90 = 3 rd angle (the right angle)<br />

<br />

<br />

<br />

<br />

<br />

<br />

25 + 65 + 90 = ? 180<br />

180 = 180 ✓<br />

25°, 65°, 90°.


a 2 + b 2 = c 2<br />

a 2 +12 2 = 13 2<br />

a 2 + 144 = 169<br />

a 2 = 25<br />

a = 25<br />

a = 5


P =2L +2W<br />

W <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

w = <br />

2w +6= <br />

P =96


P = 2L +2W<br />

96 = ? 2·34+2·14<br />

96 = 96 ✓<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

P = a + b + c,


Let x = length of 1 st side.<br />

x + 3 = length of 2 nd side<br />

2x + 2 = length of 3 rd side<br />

<br />

<br />

<br />

<br />

<br />

29 = ? 6+9+14<br />

29 = 29 ✓


w +40= <br />

<br />

<br />

<br />

<br />

<br />

P = 2L +2W<br />

360 = ? 2(110) + 2(70)<br />

360 = 360 ✓


x = <br />

<br />

a 2 + b 2 = c 2<br />

x 2 + x 2 =10 2<br />

<br />

<br />

<br />

<br />

2x 2 = 100<br />

x 2 = 50<br />

x = 50<br />

x ≈ 7.1<br />

<br />

a 2 + b 2 = c 2<br />

(7.1) 2 + (7.1) 2 ≈ 10 2 Yes.


C = πd C = πd<br />

<br />

π.<br />

V = LWH<br />

<br />

V = LWH<br />

<br />

<br />

A = 1 2 d 1 d 2 d 1 .<br />

A = 1 2 bh<br />

<br />

<br />

A = 1 2 d 1 d 2 d 2 .<br />

A = 1 2 bh<br />

<br />

<br />

A = 1 2 h⎛ ⎝b 1 + b 2<br />

⎞<br />

⎠ b 1 .<br />

<br />

A = 1 2 h⎛ ⎝b 1 + b 2<br />

⎞<br />

⎠ b 2 .<br />

<br />

h =54t + 1 2 at2 <br />

<br />

h =48t + 1 2 at2 <br />

180 = a + b + c 180 = a + b + c <br />

A = 1 pl + B <br />

2<br />

<br />

<br />

A = 1 pl + B <br />

2<br />

<br />

P =2L +2W <br />

<br />

P =2L +2W <br />

<br />

<br />

8x + y =15 <br />

<br />

9x + y =13 <br />

<br />

−4x + y =−6 <br />

<br />

−5x + y =−1 <br />

<br />

x − y =−4 <br />

<br />

x − y =−3 <br />

<br />

4x +3y =7 <br />

<br />

3x +2y =11 <br />

<br />

2x +3y =12 <br />

<br />

5x +2y =10 <br />

<br />

3x −2y =18 <br />

<br />

4x −3y =12 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

20°


C = 5 (F −32) <br />

9<br />

<br />

<br />

<br />

<br />

F = 9 C +32 <br />

5


a <br />

<br />

b <br />

<br />

<br />

c <br />

<br />

<br />

a <br />

b


0.25x + 0.10(x + 4).<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

number · value = total value<br />

23 · $0.10 = $2.30<br />

<br />

<br />

<br />

number · value = total value


p = <br />

8p +3= <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

$3.02?<br />

7(0.01) + 59(0.05) = ? 3.02<br />

3.02 = 3.02 ✓


3x −5= <br />

<br />

<br />

<br />

<br />

10(0.35) + 25(0.49) = ? 15.75<br />

3.50 + 12.25 = ? 15.75<br />

15.75 = 15.75✓


40 − f = <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

26 · 32 = 832<br />

14 · 26 = —— 364<br />

1,196 ✓


x = <br />

25 − x = <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

4.5(10) + 8(15) = ? 25(6.60)<br />

45 + 120 = ? 165<br />

165 = 165 ✓


D = rt,


+7<br />

r + 7 = Dennis’ speed<br />

r = Wayne’s speed<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

Dennis 28 mph(1.5 hours) = 42 miles<br />

Wayne 21 mph(2 hours) = 42 miles ✓


+ 15.


+ 15.<br />

<br />

<br />

<br />

Carina drove 65 mph(3 hours) = 195 miles<br />

Her brother drove 50 mph(4 hours) = ————— 200 miles<br />

395 miles ✓


Now solve this equation 70t +60t = 325<br />

130t = 325<br />

t = 2.5<br />

2.5 <br />

<br />

Truck going West 70 mph(2.5 hours) = 175 miles<br />

Truck going East 60 mph(2.5hours) = ————— 150 miles<br />

325 miles ✓


+3. <br />

<br />

<br />

30 minutes is 30<br />

60 or 1 2 hour<br />

15 minutes is 15<br />

60 or 1 4 hour


Walk 3 mph (0.5 hour) = 1.5 miles<br />

Bike 6 mph (0.25 hour) = 1.5 miles


Run 10 mph (1.5 hours) = 15 mi<br />

Bike 16 mph (2.25 hours) = 36 —— mi<br />

51 mi


a <br />

<br />

b <br />

<br />

c <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b <br />

<br />

c <br />

<br />

d <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


15 > x.<br />

<br />

<br />

<br />

<br />

<br />

x >3 <br />

x >3.<br />

x >3 <br />

<br />

<br />

<br />

x >3 (3, ∞). ∞ <br />

<br />

<br />

<br />

x > 3 <br />

<br />

<br />

x ≤1 <br />

x =1. <br />

<br />

x ≤1 (−∞, 1]. −∞ <br />

<br />

<br />

<br />

x ≤1


a x ≥−3b x < 2.5 c x ≤− 3 5<br />

a<br />

<br />

−3, −3.<br />

<br />

b<br />

<br />

<br />

c<br />

− 3 5 , −3 5 .<br />

<br />

<br />

a x > 2 b x ≤ −1.5 c x ≥ 3 4 .<br />

<br />

a x ≤ −4 b x ≥ 0.5 c x < − 2 3 .


2.5, 3, 3 2 ,4,4,99? <br />

3<br />

2


ax + b < c, ax + b ≤ c,<br />

ax + b > c, ax + b ≥ c.<br />

<br />

<br />

a ≠0<br />

ax + b < c, ax + b ≤ c, ax + b > c, ax + b ≥ c.<br />

<br />

<br />

<br />

<br />

<br />

<br />

a < b, then<br />

a + c < b + c<br />

a + c > b + c<br />

a− c < b − c<br />

a− c > b − c


multiply or divide by a positive<br />

if a < b and c > 0, then ac < bc and a c < b c .<br />

if a > b and c > 0, then ac > bc and a c > b c .<br />

multiply or divide by a negative<br />

if a < b and c < 0, then ac > bc and a c > b c .<br />

if a > b and c < 0, then ac < bc and a c < b c .<br />

<br />

<br />

<br />

<br />

<br />

x > a has the same meaning as a < x<br />

<br />

<br />

<br />

a x − 3 8 ≤ 3 4 b 9y


c<br />

5 3 .<br />

5 3 <br />

<br />

<br />

<br />

<br />

<br />

<br />

a p − 3 4 ≥ 1 6 b 9c > 72 c 24 ≤ 3 8 m<br />

<br />

<br />

a r − 1 3 ≤ 7 12 b 12d ≤ 60 c −24 < 4 3 n<br />

<br />

<br />

<br />

a −13m ≥65b<br />

−2 n ≥8<br />

a<br />

<br />

−13.<br />

−13 <br />

<br />

<br />

<br />

b


−2.<br />

−2 <br />

<br />

<br />

<br />

<br />

<br />

a −8q < 32 b<br />

k<br />

−12 ≤ 15.<br />

<br />

<br />

a −7r ≤ − 70 b u −4 ≥ −16.<br />

<br />

<br />

<br />

6y ≤11y + 17, <br />

<br />

11y <br />

<br />

<br />

−5,<br />

<br />

<br />

<br />

<br />

<br />

<br />

3q ≥ 7q − 23.


6x < 10x + 19.<br />

<br />

<br />

<br />

<br />

8p +3 ⎛ ⎝p −12 ⎞ ⎠ >7p − 28, <br />

<br />

<br />

<br />

8p +3(p −12)>7p −28<br />

<br />

<br />

7p <br />

11 > 7.<br />

<br />

<br />

<br />

8p +3p −36>7p −28<br />

11p −36>7p −28<br />

11p −36−7p >7p −28−7p<br />

4p − 36 > −28<br />

4p −36+36>−28+36<br />

4p >8<br />

<br />

<br />

4p<br />

4 > 8 4<br />

p >2<br />

<br />

(2, ∞)<br />

<br />

9y + 2 ⎛<br />

⎝y<br />

+ 6 ⎞ ⎠ > 5y − 24 <br />

<br />

<br />

6u + 8(u − 1) > 10u + 32


8x −2(5−x)


1 4 x − 12 1 x > 1 6 x + 7 <br />

8<br />

<br />

<br />

2 5 z − 1 3 z < 15 1 z − 3 <br />

5<br />

<br />

<br />

<br />

<br />

<br />

> ≥ < ≤<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

Twenty-seven less than x is at least 48.


Let n = the number of tablets.<br />

<br />

<br />

<br />

254.12n ≤ 4000<br />

<br />

<br />

n ≤ 15.74<br />

n ≤15


Let t = the number of text messages.<br />

<br />

<br />

28.80 + 0.20t ≤50<br />

0.2t ≤ 21.2<br />

t ≤ 106 text messages<br />

<br />

<br />

Yes, 28.80 + 0.20(106) = 50.


Let j = the number of invitations.<br />

<br />

<br />

2.50j − 650 ≥ 2,800<br />

2.5j ≥ 3,450<br />

j ≥ 1,380 invitations


Let h = the number of hours.<br />

<br />

<br />

<br />

525 + 780 + 95(6) ≤ 840 + 45h<br />

1,875 ≤ 840 + 45h<br />

1,035 ≤ 45h<br />

23 ≤ h<br />

h ≥23<br />

<br />

<br />

<br />

<br />

<br />

1,875 ≤ 840 + 45h<br />

1,875 ≤ 840 + 45(23)<br />

1,875 ≤ 1875


a x 3<br />

b x ≤ −0.5<br />

c x ≥ 1 3<br />

<br />

a x ≥−4<br />

b x < 2.5<br />

c x >− 3 2<br />

<br />

a x ≤5<br />

b x ≥ −1.5<br />

c x


6h −4(h −1) ≤7h −11 4k − (k −2) ≥7k −26 <br />

8m −2(14 − m) ≥7(m −4) +3m<br />

<br />

6n −12(3−n) ≤9(n −4) +9n<br />

3<br />

4<br />

b − 1 3 b < 5 12 b − 1 2<br />

<br />

9u +5(2u −5) ≥12(u −1) +7u<br />

2<br />

3<br />

g − 1 2 ⎛ ⎝g −14 ⎞ ⎠ ≤ 1 6 ⎛ ⎝g +42 ⎞ ⎠<br />

<br />

4<br />

5 h − 2 3 (h −9) ≥ 1 15 (2h +90) 5<br />

6<br />

a − 1 4 a > 7 12 a + 2 3<br />

<br />

12v +3(4v −1) ≤19(v −2) +5v<br />

<br />

15k ≤ −40 35k ≥ −77 <br />

23p −2 ⎛ ⎝6−5p ⎞ ⎠ >3 ⎛ ⎝11p −4 ⎞ ⎠<br />

<br />

18q −4 ⎛ ⎝10 − 3q ⎞ ⎠ −61 m<br />

18<br />

≥−4<br />

<br />

n<br />

13<br />

≤−6<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

−22.<br />

<br />

<br />

<br />

−7.


−5x > 10.<br />

<br />

n −3 < 12.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


2 (x + 10).<br />

5<br />

<br />

−(x − 4).<br />

<br />

<br />

<br />

<br />

<br />

x + 3 > −4 and 4x −5≤3<br />

2(y +1)−3 x ≤2,


−3 2 <br />

<br />

−3 <br />

<br />

<br />

<br />

<br />

6x −3


3x − 4 < 5 <br />

4x + 9 ≥ 1.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

3(2x +5) ≤18 2(x −7)


1 x −4≥−2 −2(x −3) ≥4. <br />

3<br />

<br />

1<br />

3 x −4≥−2 −2(x −3) ≥4<br />

1<br />

3<br />

x −4≥−2 −2x +6≥4<br />

<br />

1<br />

3 x ≥2 −2x ≥−2<br />

x ≥6 x ≤1<br />

<br />

<br />

<br />

<br />

<br />

<br />

1<br />

4<br />

x − 3 ≥ −1 <br />

−3(x − 2) ≥ 2.<br />

<br />

1<br />

5<br />

x − 5 ≥ −3 <br />

−4(x − 1) ≥ −2.<br />

a < x x < b <br />

a < x < b <br />

<br />

a < x < b. a < x x < b.<br />

Other forms:<br />

a < x < b is equivalent to a < x and x < b<br />

a ≤ x ≤ b is equivalent to a ≤ x and x ≤ b<br />

a > x > b is equivalent to a > x and x > b<br />

a ≥ x ≥ b is equivalent to a ≥ x and x ≥ b


−4≤3x −7


5−3x ≤−1 8+2x ≤5<br />

<br />

<br />

5−3x ≤−1<br />

−3x ≤−6<br />

8+2x ≤5<br />

2x ≤−3<br />

x ≥2 x ≤− 3 2<br />

<br />

<br />

<br />

<br />

<br />

⎛<br />

⎝ −∞, − 3 ⎤<br />

2⎦ ∪ [2, ∞)<br />

<br />

1 − 2x ≤ −3 <br />

7 + 3x ≤ 4.<br />

<br />

2 − 5x ≤ −3 <br />

5 + 2x ≤ 3.<br />

<br />

2 3 x −4≤3 1 (x +8) ≥−1. <br />

4<br />

<br />

2<br />

3 x −4≤3 1<br />

4<br />

(x +8) ≥−1<br />

<br />

<br />

3 ⎛ ⎝ 2 3 x −4⎞ ⎠ ≤3(3) 4·14 (x +8) ≥4·(−1)<br />

2x −12≤9<br />

2x ≤21<br />

x +8≥−4<br />

x ≥ −12<br />

x ≤ 21 2


x ≤ 21 2<br />

<br />

x ≥ −12<br />

<br />

<br />

<br />

<br />

<br />

<br />

(−∞, ∞)<br />

<br />

3<br />

5<br />

x − 7 ≤ −1 <br />

1(x + 6) ≥ −2.<br />

3<br />

<br />

3<br />

4<br />

x − 3 ≤ 3 <br />

2(x + 10) ≥ 0.<br />

5


x = <br />

24.72 + 1.54x.


x −2 x ≥−4 x ≤−1<br />

x >−6 x


1x −3≤4 <br />

3x +2≤−1 <br />

2 4<br />

1<br />

3 (x −6) ≥−2 1(x +8) ≥−3<br />

2<br />

<br />

<br />

3x +7≤1 <br />

2x +3≥−5<br />

6(2x −1) >6 <br />

5(x +2) ≥0<br />

4−7x ≥−3 <br />

5(x −3) +8>3<br />

1x −5≤3 <br />

−5≤2x −1


2−7x ≥−5 4(x −3) +7>3.<br />

<br />

a <br />

b


−|7|.<br />

<br />

, = <br />

a |−8|___ − |−8| b 12___ − |−12| c |−6|___ − 6 d −(−15)___ − |−15|<br />

<br />

14 − 2|8 − 3(4 − 1)|.<br />

<br />

<br />

<br />

<br />

<br />

|n| |n| ≥0 <br />

<br />

<br />

<br />

−5 |−5| =5.<br />

5 |5| =5.<br />

<br />

<br />

<br />

−5 <br />

|x| =5, <br />

−5


If |x| =5<br />

then x =−5 or x =5<br />

x =±5. <br />

<br />

<br />

<br />

if<br />

then<br />

|u| = a<br />

u =−a or u = a<br />

<br />

<br />

a |x| =8b |y| =−6c |z| =0<br />

a<br />

<br />

b<br />

|x| =8<br />

x =−8or x =8<br />

x =±8<br />

|y| =−6<br />

<br />

<br />

c<br />

|z| =0<br />

z =−0orz =0<br />

z =0<br />

<br />

a |x| = 2 b |y| = −4 c |z| = 0<br />

a |x| = 11 b |y| = −5 c |z| = 0


|5x −4| −3=8.<br />

<br />

|3x − 5| − 1 = 6.<br />

|4x − 3| − 5 = 2.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

2|x −7| +5=9.


2|x −7| +5=9<br />

2|x −7| =4<br />

|x −7| =2<br />

x −7=−2 x −7=2<br />

x =5 x =9<br />

<br />

3|x − 4| − 4 = 8.<br />

2|x − 5| + 3 = 9.<br />

<br />

<br />

<br />

| 2 3 x −4 | +11=3.<br />

<br />

| 2 x −4<br />

3 | +11=3<br />

<br />

| 2 3 x −4 | =−8<br />

<br />

<br />

<br />

<br />

| 3 4 x − 5 |<br />

+ 9 = 4.<br />

<br />

<br />

| 5 6 x + 3 |<br />

+ 8 = 6.<br />

|u| = |v| <br />

|x −3| = |2x +1|.<br />

<br />

<br />

|u| = a, u =−a u = a.


if |u| = |v|<br />

then u =−v or u = v<br />

<br />

<br />

<br />

if |u| = |v|<br />

then u =−v or u = v<br />

<br />

<br />

|5x −1| = |2x +3|.<br />

<br />

|5x −1| = |2x +3|<br />

<br />

<br />

5x −1 = −(2x +3) or 5x −1 = 2x +3<br />

5x −1 = −2x − 3 or 3x −1 = 3<br />

7x −1 = −3 3x = 4<br />

7x = −2 x = 4 3<br />

x = − 2 7<br />

or x = 4 3<br />

<br />

<br />

|7x − 3| = |3x + 7|.<br />

|6x − 5| = |3x +4|.<br />

<br />

<br />

<br />

<br />

|x| =5, −5 <br />

<br />

|x| =5<br />

x =−5 or x =5<br />

|x| ≤5? −5 <br />

−5


|u| ≤ a, −a ≤ u ≤ a. <br />

<br />

<br />

< ≤<br />

<br />

if |u| < a, then −a < u < a<br />

if |u| ≤ a, then −a ≤ u ≤ a<br />

<br />

<br />

<br />

<br />

|x|


|5x −6| ≤4<br />

−4 ≤ 5x −6≤4<br />

2≤5x ≤10<br />

2<br />

5 ≤ x ≤2<br />

<br />

<br />

⎡<br />

⎣<br />

2<br />

5<br />

,2 ⎤ ⎦<br />

<br />

<br />

<br />

|2x − 1| ≤ 5. <br />

<br />

|4x − 5| ≤ 3. <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

|u| < a is equivalent to −a < u < a<br />

|u| ≤ a is equivalent to −a ≤ u ≤ a<br />

<br />

<br />

<br />

|x| ≤5. <br />

−5 −5 <br />

<br />

|x| ≥5. <br />

<br />

−5


−5 <br />

<br />

|u| ≥ a, u ≤−a u ≤ a. <br />

<br />

<br />

<br />

<br />

<br />

if |u| > a, then u a<br />

if |u| ≥ a, then u ≤−a or u ≥ a<br />

|x| >4. <br />

<br />

|x| >4<br />

x 4<br />

<br />

<br />

<br />

(−∞, −4) ∪ (4, ∞)<br />

<br />

−6, <br />

<br />

|x| > 2. <br />

<br />

|x| > 1.


|2x −3| ≥5. <br />

<br />

|2x −3| ≥5<br />

<br />

<br />

<br />

<br />

2x −3≤−5 or 2x −3≥5<br />

2x ≤−2 or 2x ≥8<br />

x ≤−1 or x ≥4<br />

<br />

(−∞, −1] ∪ [4, ∞)<br />

<br />

<br />

<br />

|4x −3| ≥ 5. <br />

<br />

|3x − 4| ≥ 2. <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

|u| > a is equivalent to u < − a or u > a<br />

|u| ≥ a is equivalent to u ≤ −a or u ≥ a<br />

<br />

<br />

<br />

<br />

<br />

|actual-ideal| ≤ tolerance<br />

<br />

<br />

0.075


|actual-ideal| ≤ tolerance<br />

|x −60| ≤ 0.075<br />

−0.075 ≤ x − 60 ≤ 0.075<br />

59.925 ≤ x ≤ 60.075


a |x| =6 b |y| =−3 c<br />

|z| =0<br />

a |x| =4 b |y| =−5 c<br />

|z| =0<br />

a |x| =7 b |y| = −11 c<br />

|z| =0<br />

a |x| =3 b |y| =−1 c<br />

|z| =0<br />

|2x −3| −4=1 |4x −1| −3=0<br />

|3x −4| +5=7 |4x +7| +2=5 4|x −1| +2=10<br />

3|x −4| +2=11 3|4x −5| −4=11 3|x +2| −5=4<br />

−2|x −3| +8=−4<br />

−3|x −4| +4=−5<br />

<br />

| 3 4 x −3 | +7=2<br />

<br />

| 3 5 x −2 | +5=2 | 1 2 x +5 | +4=1 | 1 4 x +3 | +3=1<br />

|3x −2| = |2x −3| |4x +3| = |2x +1| |6x −5| = |2x +3|<br />

|6−x| = |3−2x|<br />

<br />

<br />

|x| 5 |x +3| ≥5<br />

|x −7| ≥1 3|x| +4≥1 5|x| +6≥1<br />

<br />

2|x +6| +4=8 |6x −5| = |2x +3| |3x −4| ≥2


|2x −5| +2=3 |4x −3|


a ≠0, ax + b =0.


change = new amount − original amount<br />

<br />

<br />

change is what percent of the original amount?<br />

<br />

<br />

<br />

<br />

amount of discount = discount rate · original price<br />

sale price = original amount − discount<br />

<br />

amount of mark-up = mark-up rate · original cost<br />

list price = original cost + mark up<br />

<br />

<br />

<br />

I = interest<br />

I = Prt where P = principal<br />

r = rate<br />

t = time<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

number · value = total value


x > a x ≥ a x < a x ≤ a<br />

<br />

<br />

<br />

<br />

<br />

a ≠0:<br />

ax + b < c, ax + b ≤ c, ax + b > c, ax + b ≥ c.<br />

<br />

a < b, then<br />

<br />

a + c < b + c<br />

a + c > b + c<br />

a− c < b − c<br />

a− c > b − c<br />

<br />

<br />

<br />

multiply or divide by a positive<br />

if a < b and c > 0, then ac < bc and a c < b c .<br />

if a > b and c > 0, then ac > bc and a c > b c .<br />

multiply or divide by a negative<br />

if a < b and c < 0, then ac > bc and a c > b c .<br />

if a > b and c < 0, then ac < bc and a c < b c .<br />

<br />

<br />

> ≥ < ≤


a < x < b a < x x < b.<br />

a < x < b is equivalent to a < x and x < b<br />

a ≤ x ≤ b is equivalent to a ≤ x and x ≤ b<br />

Other forms:<br />

a > x > b is equivalent to a > x and x > b<br />

a ≥ x ≥ b is equivalent to a ≥ x and x ≥ b<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

|n| |n| ≥0 <br />

<br />

<br />

<br />

if |u| = a<br />

then u =−a or u = a<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

if |u| = |v|<br />

then u =−v or u = v<br />

< ≤<br />

<br />

if |u| < a, then −a < u < a<br />

if |u| ≤ a, then −a ≤ u ≤ a<br />

< ≤<br />

<br />

<br />

|u| < a is equivalent to −a < u < a<br />

|u| ≤ a is equivalent to −a ≤ u ≤ a<br />

<br />

<br />

<br />

> ≥<br />

<br />

if |u| > a, then u a<br />

if |u| ≥ a, then u ≤−a or u ≥ a<br />

> ≥


|u| > a is equivalent to u a<br />

|u| ≥ a is equivalent to u ≤−a or u ≥ a<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

10x −1=5x, x = 1 5<br />

−12n +5=8n, n =− 5 4<br />

<br />

6(x +6) =24 −(s +4) =18<br />

23 − 3 ⎛ ⎝y −7 ⎞ ⎠ =8 1<br />

3<br />

(6m +21) = m −7<br />

4 ⎛ ⎝3.5y + 0.25 ⎞ ⎠ = 365 0.25 ⎛ ⎝q −8 ⎞ ⎠ = 0.1 ⎛ ⎝q +7 ⎞ ⎠<br />

8(r −2) =6(r +10) 5+7(2−5x) =2(9x +1) − (13x −57)<br />

(9n +5) − (3n −7) =20−(4n −2) 2 ⎡ ⎣−16 + 5(8k −6) ⎤ ⎦ =8(3−4k) −32<br />

<br />

<br />

<br />

17y −3 ⎛ ⎝4−2y ⎞ ⎠ =11 ⎛ ⎝y −1 ⎞ ⎠ +12y −1 9u +32=15(u −4) −3(2u +21)<br />

−8(7m +4) =−6(8m +9)<br />

<br />

<br />

2<br />

5<br />

n − 1 10 = 7 10<br />

3<br />

4<br />

a − 1 3 = 1 2 a + 5 6<br />

1<br />

2<br />

(k +3) = 1 3 (k +16) <br />

5y −1<br />

3<br />

+4=<br />

−8y +4<br />

6<br />

0.8x − 0.3 = 0.7x + 0.2 0.10d + 0.05(d −4) = 2.05


−135. <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


V = LWH <br />

A = 1 2 d 1 d 2 d 2 . h =48t + 1 2 at2 <br />

<br />

4x −3y =12 <br />

<br />

<br />

<br />

67.5 <br />

<br />

<br />

45°


x 2 −2 < x 15p −30 9h −7(h −1)≤4h −23<br />

<br />

5n − 15(4 − n) < 10(n −6)+10n<br />

3<br />

8<br />

a − 1 12 a > 5 12 a + 3 4


x ≤5 x >−3 4x −2≤4 <br />

5(3x −2) ≤5 <br />

7x −1>−8<br />

4(x +2) 3<br />

4<br />

2(x +3) ≥0 <br />

3(x +4) ≤6<br />

<br />

1<br />

2<br />

x −3≤4 <br />

1(x −6) ≥−2<br />

3<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

|x| =8 |y| = −14 |z| =0<br />

|3x −4| +5=7 4|x −1| +2=10 −2|x −3| +8=−4


| 1 2 x +5 | +4=1 |6x −5| = |2x +3|<br />

<br />

<br />

|x| ≤8 |2x −5| ≤3 |6x −5| 6 |x| ≥2 |x −5| >−2<br />

|x −7| ≥1 3|x| +4≥1


−5(2x +1) =45 <br />

<br />

1<br />

4 (12m +28) =6+2(3m +1) 8(3a +5) −7(4a −3) =20−3a<br />

0.1d + 0.25(d +8) = 4.1 <br />

14n −3(4n +5) =−9+2(n −8)<br />

<br />

3(3u +2) +4 ⎡ ⎣6−8(u −1) ⎤ ⎦ =3(u −2)<br />

3<br />

4<br />

x − 2 3 = 1 2 x + 5 6<br />

|3x −4| =8 |2x −1| = |4x +3|<br />

<br />

x +2y =5 <br />

<br />

x ≥ −3.5 x < 11 −2 ≤ x


4 1 <br />

2


5x − 4 x = −1.<br />

<br />

3x − 2y x = 4, y = −3.<br />

<br />

8 − 3y = 20.


(x, y) <br />

<br />

<br />

(0, 0). (0, 0) <br />

<br />

(0, 0) <br />

(1, 3) <br />

x =1. <br />

y =3. (1, 3).


x =1 y =3 <br />

(1, 3).<br />

(0, 4) <br />

(−2, 0) <br />

<br />

<br />

(a, 0).<br />

(0, b).<br />

<br />

<br />

a (−5, 4) b (−3, −4) c (2, −3) d (0, −1) e ⎛ ⎝ 3, 5 ⎞<br />

2⎠ .<br />

<br />

<br />

<br />

<br />

a x =−5, y =4, <br />

(−5, 4) <br />

b x =−3, y =−4,


(−3, −4) <br />

c x =2, y =−3, (2, −3)<br />

<br />

d x =0, (0, −1) <br />

e x =3, y = 5 , <br />

2<br />

5 2 ⎛ ⎝ 3, 5 ⎞<br />

2⎠ <br />

<br />

<br />

a (−2, 1) b (−3, −1) c (4, −4) d (−4, 4) e ⎛ ⎝ ⎛ −4, 3<br />

2<br />

⎞<br />

⎠<br />

⎞<br />

<br />

<br />

a (−4, 1) b (−2, 3) c (2, −5) d (−2, 5) e ⎛ ⎝ ⎛ −3, 5<br />

2<br />

⎞<br />

⎠<br />

⎞<br />

<br />

<br />

<br />

Quadrant I Quadrant II Quadrant III Quadrant IV<br />

⎛<br />

⎝ x, y ⎞ ⎠<br />

⎛<br />

⎝ x, y ⎞ ⎠<br />

⎛<br />

⎝ x, y ⎞ ⎠<br />

⎛<br />

⎝ x, y ⎞ ⎠<br />

⎛<br />

⎝ +, + ⎞ ⎠<br />

⎛<br />

⎝ −, + ⎞ ⎠<br />

⎛<br />

⎝ −, − ⎞ ⎠<br />

⎛<br />

⎝ +, − ⎞ ⎠


Ax + By = C. <br />

<br />

Ax + By = C, <br />

<br />

y =−3x +5 Ax + By = C. <br />

Ax + By = C <br />

y = −3x +5<br />

Add to both sides. y +3x = −3x +5+3x<br />

Simplify. y +3x = 5<br />

Use the Commutative Property to put it in<br />

Ax + By = C form. 3x + y = 5<br />

y =−3x +5 3x + y =5, <br />

Ax + By = C. Ax + By = C, <br />

<br />

<br />

Ax + By = C.<br />

A ≥0 <br />

<br />

<br />

(x, y). <br />

<br />

<br />

<br />

(x, y) Ax + By = C,


Ax + By = C <br />

<br />

<br />

<br />

<br />

<br />

y =2x −3 <br />

<br />

a <br />

b <br />

(0, −3) (3, 3) (2, −3) (−1, −5)<br />

<br />

<br />

a<br />

b (0, −3), (3, 3), (2, −3), (−1, −5).


(0, 3), (3, −3), (−1, −5) y =2x −3, (2, −3) <br />

y =2x −3 <br />

<br />

y = 3x − 1. <br />

a <br />

b <br />

(0, −1) (2, 5)


y = 3x − 1. <br />

a <br />

b <br />

(3, −1) (−1, −4)<br />

<br />

<br />

<br />

<br />

<br />

<br />

y =2x +1


y = 2x − 3.<br />

y = −2x + 4.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

x <br />

<br />

<br />

<br />

y = 1 2 x +3.<br />

<br />

1 2


1 2 <br />

<br />

y = 1 2 x +3<br />

(x, y)<br />

(0, 3)<br />

(2, 4)<br />

(4, 5)<br />

<br />

<br />

<br />

y = 1 3 x − 1.<br />

<br />

y = 1 4 x + 2.<br />

<br />

<br />

<br />

x =−3.


−3, −3.<br />

−3 <br />

<br />

<br />

x =−3<br />

(x, y)<br />

−3 (−3, 1)<br />

−3 (−3, 2)<br />

−3 (−3, 3)<br />

<br />

<br />

y =4. <br />

<br />

<br />

y =4<br />

(x, y)<br />

(0, 4)<br />

(2, 4)<br />

(4, 4)


x = a.<br />

(a, 0).<br />

y = b.<br />

(0, b).<br />

<br />

a x =2b y =−1.<br />

<br />

a <br />

<br />

x =2<br />

(x, y)<br />

(2, 1)<br />

(2, 2)<br />

(2, 3)


y =−1 <br />

−1.<br />

y =−1<br />

(x, y)<br />

−1 (0, −1)<br />

−1 (3, −1)<br />

−3 −1 (−3, −1)<br />

a x = 5 b y = −4.<br />

a x = −2 b y = 3.<br />

y =4x y =4?<br />

y =4x


y =4 <br />

<br />

y =4x y =4 <br />

<br />

y =−3x y =−3


y = −4x y = −4.<br />

y = 3 y = 3x.


(3, 0) (0, 6)<br />

(4, 0) −3 (0, −3)<br />

(5, 0) −5 (0, 5)<br />

(0, 0) (0, 0)<br />

(a, 0) (0, b)<br />

<br />

<br />

<br />

(a, 0) <br />

<br />

(0, b) <br />

<br />

(a, 0) <br />

(0, b)


a (4, 0). (4, 0).<br />

(0, 2). (0, 2).<br />

b (2, 0). (2, 0).<br />

(0, −6). (0, −6).<br />

c (−5, 0). (−5, 0).<br />

(0, −5). (0, −5).


y =0 <br />

x =0 <br />

<br />

<br />

<br />

<br />

y =0 <br />

x =0 <br />

<br />

2x + y =8.<br />

<br />

y =0 x =0


y =0.<br />

y =0.<br />

<br />

(4, 0)<br />

x =0.<br />

x =0.<br />

<br />

(0, 8)<br />

(4, 0) (0, 8) <br />

2x + y =8<br />

<br />

<br />

<br />

<br />

<br />

<br />

3x + y = 12.<br />

x + 4y = 8.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

– x +2y =6


x – 2y = 4.


– x + 3y = 6.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

y = 0 <br />

x = 0 <br />

<br />

<br />

<br />

<br />

4x −3y =12 <br />

<br />

<br />

<br />

4x −3y =12<br />

(x, y)<br />

(3, 0)<br />

−4 (0, −4)<br />

(6, 4)


5x − 2y = 10.<br />

3x − 4y = 12.


y =5x <br />

<br />

(0, 0).<br />

<br />

<br />

<br />

y =5x<br />

(x, y)<br />

(0, 0)<br />

(1, 5)<br />

−1 −5 (−1, −5)<br />

<br />

y = 4x.


y = −x.


a (−4, 2) b (−1, −2) c (3, −5) d (−3, 0) e<br />

⎛<br />

⎝ 5 3 ,2⎞ ⎠<br />

<br />

a (−2, −3) b (3, −3) c (−4, 1) d (4, −1) e<br />

⎛<br />

⎝ 3 2 ,1⎞ ⎠<br />

a (3, −1) b (−3, 1) c (−2, 0) d (−4, −3) e<br />

⎛<br />

⎝ 1, 14 ⎞<br />

5 ⎠<br />

a (−1, 1) b (−2, −1) c (2, 0) d (1, −4) e<br />

⎛<br />

⎝ 3, 7 ⎞<br />

2⎠<br />

<br />

a b <br />

y = x +2;<br />

y = x −4;<br />

(0, 2); (1, 2); (−1, 1); (−3, −1).<br />

(0, −4); (3, −1); (2, 2); (1, −5).


y = 1 2 x −3;<br />

(0, −3); (2, −2); (−2, −4); (4, 1)<br />

y = 1 3 x +2;<br />

(0, 2); (3, 3); (−3, 2); (−6, 0).<br />

<br />

<br />

y = x +2 y = x −3 y =3x −1<br />

y =−2x +2 y =−x −3 y =−x −2<br />

y =2x y =−2x y = 1 2 x +2<br />

y = 1 3 x −1 y = 4 3 x −5 y = 3 2 x −3<br />

y =− 2 5 x +1 y =−4 5 x −1 y =−3 2 x +2<br />

y =− 5 3 x +4<br />

<br />

<br />

a x =4b y =3 a x =3b y =1 a x =−2b y =−5<br />

a x =−5b y =−2<br />

<br />

y =2x y =2 y =5x y =5 y =− 1 2 x y =−1 2<br />

y =− 1 3 x y =−1 3


x − y =5 x − y =−4 3x + y =6<br />

x −2y =8 4x − y =8 5x − y =5<br />

2x +5y =10 3x −2y =12<br />

<br />

<br />

−x +4y =8 x +2y =4 x + y =−3<br />

x − y =−4 4x + y =4 3x + y =3<br />

3x − y =−6 2x − y =−8 2x +4y =12<br />

3x −2y =6 2x −5y = −20 3x −4y = −12<br />

y =−2x y =5x y = x


y =−x<br />

<br />

<br />

y = 3 2 x y =−2 3 x y =−1 2 x +3<br />

y = 1 4 x −2 4x + y =2 5x +2y =10<br />

y =−1 x =3<br />

<br />

<br />

y = 1 5 x −2.<br />

<br />

<br />

<br />

<br />

4x + y =−4? <br />

<br />

<br />

y = 2 x −2? <br />

3<br />

<br />

a <br />

b


(1 – 4)<br />

<br />

(8 − 2) .<br />

<br />

<br />

0<br />

4<br />

, 4 0 .<br />

<br />

<br />

15<br />

−3 , −15<br />

3 , −15<br />

−3 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

m = rise run .


m = rise run .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

m = rise run .<br />

<br />

<br />

<br />

<br />

<br />

(0, 5) (3, 3)<br />

(0, 5), <br />

(3, 3) <br />

−2.<br />

<br />

<br />

<br />

<br />

m = rise run<br />

m = −2<br />

3<br />

m =− 2 3<br />

− 2 3 .


y =4, <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

m = rise run<br />

m = 0 3<br />

m =0<br />

y = 4 is 0.<br />

x =3,


m = rise run<br />

m = 2 0<br />

x =3 <br />

<br />

<br />

<br />

y = b, <br />

x = a, <br />

<br />

a x =8b y =−5.<br />

<br />

a x =8<br />

<br />

b y =−5<br />

<br />

x = −4.<br />

y = 7.<br />

<br />

<br />

<br />

<br />

(x, y)


(x, y) <br />

<br />

(x 1 , y 1 ) read “x sub 1, y sub 1”<br />

(x 2 , y 2 ) read “x sub 2, y sub 2”<br />

(x 1 , y 1 ) (x 2 , y 2 ) <br />

(x 3 , y 3 ), (x 4 , y 4 ), <br />

<br />

(2, 3) (7, 6), <br />

<br />

x<br />

⎛ 1 , y 1 x<br />

⎝ 2, ⎞⎛ 2 , y 2 ⎞<br />

3 ⎠⎝ 7, 6 ⎠<br />

m = rise run<br />

m = 3 5<br />

<br />

<br />

<br />

<br />

y 2 , y 1 , <br />

<br />

<br />

<br />

m = 6−3<br />

7−2<br />

m = y 2 − y 1<br />

7−2<br />

m = y 2 − y 1<br />

x 2 − x 1<br />

m = y 2 − y 1<br />

x 2 − x m = rise<br />

1 run . <br />

<br />

<br />

(x 1 , y 1 ) (x 2 , y 2 ) <br />

m = y 2 − y 1<br />

x 2 − x 1<br />

.<br />

<br />

y of the second point minus y of the first point<br />

over<br />

x of the second point minus x of the first point.


(−2, −3) (−7, 4).<br />

<br />

(−2, −3) (−7, 4) <br />

⎛ x 1 , y 1 ⎞<br />

⎝−2, −3⎠ ⎛ x 2 , y 2 ⎞<br />

⎝−7, 4 ⎠<br />

m = y 2 − y 1<br />

x 2 − x 1<br />

<br />

y y <br />

x x <br />

m = 4 − (−3)<br />

−7 − (−2)<br />

<br />

m = 7 −5<br />

m =− 7 5<br />

<br />

m = rise run<br />

m = 7 −5<br />

m =− 7 5<br />

<br />

(−3, 4) (2, −1).<br />

<br />

(−2, 6) (−3, −4).


(1, −1) m = 3 4 .<br />

<br />

m = 3 4 , m = −3<br />

−4 <br />

(1, −1) −3, −4.<br />

<br />

(2, −2) m = 4 3 .<br />

<br />

(−2, 3) m = 1 4 .


m = rise run <br />

<br />

<br />

<br />

<br />

<br />

<br />

y = 1 x +3 <br />

2<br />

<br />

<br />

m = rise run<br />

m = 1 2<br />

(0, 3).


y = 1 x +3 <br />

2<br />

<br />

<br />

(0, b) y = mx + b.<br />

<br />

<br />

<br />

a y =− 4 x −2 b x +3y =9<br />

7<br />

<br />

a <br />

<br />

<br />

<br />

<br />

b <br />

x +3y =9


a y = 2 5 x − 1 b x + 4y = 8<br />

<br />

<br />

a y = − 4 x + 1 b 3x + 2y = 12<br />

3<br />

<br />

<br />

<br />

y =−x +4 <br />

<br />

y = mx + b<br />

y =−x +4<br />

m =−1<br />

(0, 4)<br />

<br />

<br />

<br />

<br />

m = −1<br />

1<br />

−1 <br />

<br />

<br />

y = −x − 3 <br />

<br />

y = −x − 1


Equation<br />

Method<br />

#1 x = 2 Vertical line<br />

#2 y = −1 Horizontal line<br />

#3 −x +2y = 6 Intercepts<br />

#4 4x −3y = 12 Intercepts<br />

#5 y =−x + 4 Slope–intercept<br />

<br />

<br />

<br />

<br />

Ax + By = C. y =0 x =0 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

x = a <br />

y = b <br />

<br />

<br />

y = mx + b, <br />

<br />

Ax + By = C,


a y =5b 4x −5y =20c x =−3d y =− 5 9 x +8<br />

a y =5<br />

<br />

5 <br />

b 4x −5y =20<br />

Ax + By = C. <br />

c x =−3<br />

−3.<br />

d y =− 5 9 x +8<br />

y = mx + b <br />

<br />

<br />

a 3x + 2y = 12 b y = 4 c y = 1 5 x − 4 d x = −7.<br />

<br />

<br />

a x = 6 b y = − 3 x + 1 c y = −8 d 4x − 3y = −1.<br />

4<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

F = 9 C +32 <br />

5<br />

<br />

a <br />

b <br />

c <br />

d <br />

a<br />

<br />

F = 9 5 C +32<br />

F C =0. F = 9 (0) +32<br />

5<br />

F =32


F = 9 5 C +32<br />

F C = 20. F = 9 (20) +32<br />

5<br />

F =36+32<br />

F =68<br />

c<br />

<br />

<br />

9<br />

5<br />

, <br />

<br />

0° 32° <br />

d <br />

(0, 32) <br />

<br />

<br />

h = 2s + 50 <br />

a <br />

b <br />

c <br />

d


T = 1 n + 40 <br />

<br />

4<br />

<br />

a <br />

b <br />

c <br />

<br />

d <br />

<br />

<br />

<br />

<br />

<br />

C = 0.5m +60 <br />

<br />

a <br />

b <br />

c <br />

d <br />

a<br />

<br />

C = 0.5m +60<br />

C m =0. C = 0.5(0) +60<br />

C =60<br />

<br />

b<br />

C = 0.5m +60<br />

C m = 250. C = 0.5(250) +60<br />

C = 185<br />

<br />

c


d <br />

(0, 60) <br />

<br />

m = 0.5<br />

m = 0.5<br />

1<br />

m = 0.5(100)<br />

1(100)<br />

m = 50<br />

100<br />

<br />

<br />

C = 4p + 25 <br />

<br />

<br />

a <br />

b <br />

c <br />

d <br />

<br />

C = 1.8n + 35 <br />

<br />

a <br />

b <br />

c <br />

d


m = 2 , <br />

5<br />

<br />

<br />

<br />

<br />

<br />

<br />

m 1 m 2 m 1 = m 2 .


a 3x −2y =6 y = 3 x +1 b y =2x -3 −6x +3y =−9.<br />

2<br />

a<br />

<br />

3x −2y =6<br />

y = 3 2 x +1<br />

y.<br />

−2y =−3x +6<br />

−2y<br />

= −3x +6<br />

−2 −2<br />

y = 3 2 x −3<br />

<br />

<br />

y <br />

<br />

y = 3 2 x −3<br />

y = mx + b<br />

m = 3 2<br />

y = 3 2 x +1<br />

y = 3 2 x +1<br />

y = mx + b<br />

y = 3 2<br />

y-intercept is ⎛ ⎝0, −3 ⎞ ⎠<br />

y-intercept is ⎛ ⎝0, 1 ⎞ ⎠<br />

<br />

<br />

b<br />

y =2x −3 −6x +3y =−9<br />

<br />

<br />

y<br />

=2x −3<br />

y.<br />

−6x +3y =−9<br />

3y =6x −9<br />

3y<br />

= 6x −9<br />

3 3<br />

y =2x −3<br />

<br />

<br />

y<br />

=2x −3<br />

y <br />

<br />

y =2x −3<br />

y = mx + b<br />

m =2<br />

y =2x −3<br />

y = mx + b<br />

m =2<br />

y-intercept is (0, −3) y-intercept is (0, −3)


a 2x + 5y = 5 y = − 2 5 x − 4 b y = − 1 x − 1 x + 2y = −2.<br />

2<br />

<br />

<br />

a 4x − 3y = 6 y = 4 3 x − 1 b y = 3 x − 3 3x − 4y = 12.<br />

4<br />

<br />

<br />

a y =−4 y =3b x =−2 x =−5.<br />

<br />

a y =−4 y =3<br />

<br />

y =−4 y =3, (0, −4) (0, 3).<br />

<br />

b x =−2 x =−5<br />

<br />

x =−2 x =−5, (−2, 0) (−5, 0).<br />

<br />

<br />

<br />

a y = 8 y = −6 b x = 1 x = −5.<br />

<br />

<br />

a y = 1 y = −5 b x = 8 x = −6.<br />

y = 1 x −1 y =−4x +2, <br />

4


m 1 = 1 4 , m 2 =−4, <br />

−1.<br />

m 1 · m 2<br />

1<br />

4 (−4)<br />

−1<br />

<br />

<br />

<br />

m 1 m 2 <br />

m 1 =− 1 m 2<br />

.<br />

<br />

−1 m 1 · m 2 =−1.<br />

<br />

<br />

<br />

<br />

<br />

−1, <br />

<br />

<br />

a y =−5x −4 x −5y =5b 7x +2y =3 2x +7y =5<br />

a


y =−5x −4<br />

y.<br />

x −5y =5<br />

−5y =−x +5<br />

−5y<br />

−5<br />

y<br />

=<br />

−x +5<br />

−5<br />

= 1 5 x −1<br />

<br />

y =−5x −4<br />

y = mx + b<br />

m 1 =−5<br />

y = 1 5 x −1<br />

y = mx + b<br />

m 2 = 1 5<br />

<br />

−5 ⎛ ⎝ 1 ⎞<br />

5⎠ =−1, <br />

b<br />

y.<br />

<br />

7x +2y =3<br />

2y =−7x +3<br />

2y<br />

2<br />

=<br />

−7x +3<br />

2<br />

y =− 7 2 x + 3 2<br />

y<br />

= mx + b<br />

m 1 =− 7 2<br />

2x +7y =5<br />

7y =−2x +5<br />

7y<br />

7<br />

=<br />

−2x +5<br />

7<br />

y =− 2 7 x + 5 7<br />

y<br />

= mx + b<br />

m 1 =− 2 7<br />

<br />

<br />

<br />

<br />

a y =−3x + 2 x − 3y = 4 b 5x + 4y = 1 4x + 5y = 3.<br />

<br />

<br />

a y = 2x − 5 x + 2y = −6 b 2x − 9y = 3 9x − 2y = 1.


y =3 y =−2 x =−5<br />

x =4<br />

<br />

(2, 5), (4, 0) (3, 6), (8, 0) (−3, 3), (4, −5)<br />

(−2, 4), (3, −1) (−1, −2), (2, 5) (−2, −1), (6, 5)<br />

(4, −5), (1, −2) (3, −6), (2, −2)


(2, 5); m =− 1 3<br />

(1, 4) m =− 1 2<br />

(−1, −4) m = 4 3<br />

(−3, −5) m = 3 2<br />

m =− 2 5<br />

−2 m = 3 4<br />

(−4, 2) m =4 (1, 5) m =−3<br />

<br />

<br />

y =−7x +3 y =4x −10 3x + y =5<br />

4x + y =8 6x +4y =12 8x +3y =12<br />

5x −2y =6 7x −3y =9<br />

<br />

y =3x −1 y =2x −3 y =−x +3<br />

y =−x −4 y =− 2 5 x −3 y =−3 5 x +2<br />

3x −2y =4 3x −4y =8<br />

<br />

<br />

x =2 y =5 y =−3x +4<br />

x − y =5 x − y =1 y = 2 3 x −1<br />

3x −2y = −12<br />

2x −5y = −10


P = 31 + 1.75w <br />

<br />

<br />

<br />

<br />

a <br />

<br />

<br />

b <br />

<br />

<br />

c <br />

<br />

d <br />

<br />

P = 28 + 2.54w <br />

<br />

<br />

<br />

<br />

a <br />

<br />

<br />

b <br />

<br />

<br />

c <br />

<br />

d <br />

<br />

<br />

R = 0.575m +42 <br />

<br />

<br />

<br />

<br />

a <br />

<br />

<br />

b <br />

<br />

<br />

c <br />

<br />

d <br />

<br />

<br />

C = 0.32m +15<br />

<br />

<br />

<br />

<br />

a <br />

<br />

b <br />

<br />

c <br />

<br />

d <br />

<br />

<br />

<br />

<br />

S = 400 + 0.15c <br />

<br />

<br />

<br />

a <br />

<br />

b <br />

<br />

c <br />

<br />

d <br />

<br />

<br />

<br />

S = 750 + 0.09c <br />

<br />

<br />

<br />

a <br />

<br />

b <br />

<br />

c <br />

<br />

d <br />

<br />

<br />

C = 450 + 28g <br />

<br />

<br />

<br />

a <br />

<br />

b <br />

<br />

c <br />

<br />

d <br />

<br />

<br />

C = 750 + 42g <br />

<br />

<br />

<br />

a <br />

<br />

b <br />

<br />

c <br />

<br />

d


y = 3 4 x −3; 3x −4y =−2 3x +4y =−2; y = 3 2x −4y =6; x −2y =3<br />

4 x −3<br />

<br />

8x +6y =6; 12x +9y =12<br />

x =5; x =−6 x =−3; x =−2<br />

4x −2y =5; 3x +6y =8 <br />

8x −2y =7; 3x +12y =9<br />

<br />

3x −6y = 12; 6x −3y =3<br />

<br />

9x −5y =4; 5x +9y =−1<br />

<br />

7x −4y =8; 4x +7y =14<br />

5x −2y = 11; 5x − y =7<br />

3x −2y =8; 2x +3y =6 2x +3y =5; 3x −2y =7 3x −2y =1; 2x −3y =2<br />

2x +4y =3; 6x +3y =2 y =2; y =6 y =−1; y =2<br />

<br />

m = 1 2<br />

m =2?<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


y-intercept<br />

<br />

<br />

<br />

<br />

<br />

<br />

2 (x + 15).<br />

5<br />

<br />

<br />

−3 ⎛<br />

⎝x<br />

− (−2) ⎞ ⎠.<br />

<br />

y − 3 =−2(x + 1).<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

y = mx + b.<br />

<br />

<br />

−9 (0, −4).<br />

<br />

<br />

y = mx + b.


y = mx + b.<br />

<br />

2 5<br />

(0, 4).<br />

<br />

−1 (0, −3).<br />

<br />

<br />

<br />

<br />

<br />

y = mx + b.<br />

<br />

(0, −4) (3, −2).<br />

<br />

<br />

y = mx + b.


(x 1 , y 1 ) <br />

(x, y).


m = y − y 1<br />

x − x 1<br />

x − x 1 . m(x − x 1 ) = ⎛ ⎝y − y 1<br />

x − x 1<br />

⎞<br />

⎠ (x − x 1 )<br />

m(x − x 1 ) = y − y 1<br />

y y − y 1 = m(x − x 1 )<br />

<br />

<br />

(x 1 , y 1 ) <br />

y − y 1 = m(x − x 1 )<br />

<br />

<br />

<br />

<br />

<br />

m =− 1 3<br />

(6, −4). <br />

<br />

<br />

<br />

m = − 2 5<br />

(10, −5).<br />

<br />

m = − 3 , (4, −7).<br />

4


y − y 1 = m(x − x 1 ).<br />

<br />

<br />

(−2, −6). <br />

<br />

y − y 1 = m(x − x 1 ).<br />

<br />

<br />

y − y 1 = m(x − x 1 ).<br />

<br />

y =0x −6.<br />

y = a?<br />

(−3, 8).<br />

(−1, 4).<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

(−3, −1) (2, −2)


(−2, −4) (1, −3).<br />

<br />

(−4, −3) (1, −5).<br />

<br />

<br />

<br />

<br />

<br />

<br />

m = y 2 − y 1<br />

x 2 − x 1<br />

<br />

y − y 1 = m(x − x 1 ).<br />

<br />

<br />

(−3, 5) (−3, 4).


(−3, 5) (−3, 4). m = y 2 − y 1<br />

x 2 − x 1<br />

m =<br />

m = −1<br />

0<br />

4−5<br />

−3 − (−3)<br />

<br />

−2. x =−2.<br />

<br />

<br />

<br />

<br />

(5, 1) (5, −4).<br />

(−4, 4) (−4, 3).<br />

<br />

<br />

<br />

<br />

y = mx + b<br />

y − y 1 = m(x − x 1 )<br />

y − y 1 = m(x − x 1 )<br />

<br />

<br />

<br />

<br />

<br />

y =2x −3. (−2, 1).


y =2x −3. <br />

m ∥ =2. m ∥ <br />

<br />

(−2, 1) m =2.<br />

(−2, 1) <br />

m =2 m = 2 1 <br />

(−2, 1)?<br />

<br />

<br />

<br />

<br />

<br />

y =2x −3 (−2, 1).


y = 3x + 1 (4, 2). <br />

<br />

<br />

y = 1 x − 3 (6, 4).<br />

2<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

y − y 1 = m(x − x 1 ).<br />

<br />

<br />

<br />

<br />

<br />

y =2x −3. (−2, 1).<br />

<br />

m ⊥ <br />

⊥ <br />

y =2x − 3 perpendicular line<br />

m =2 m ⊥ =− 1 2<br />

(−2, 1) m ⊥ =− 1 2 .<br />

(−2, 1) −1


(−2, 1)?<br />

<br />

<br />

<br />

<br />

<br />

y =2x −3 (−2, 1). <br />

<br />

<br />

<br />

y = 3x + 1 (4, 2).


y = 1 x − 3 (6, 4). <br />

2<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

y − y 1 = m(x − x 1 ).<br />

<br />

<br />

x =5 (3, −2). <br />

<br />

<br />

<br />

x =5. <br />

m ⊥ =0.<br />

(3, −2)<br />

m ⊥ =0<br />

y − y 1 = m(x − x 1 ). y − y 1 = m(x − x 1 )<br />

y − (−2) =0(x −3)<br />

y +2=0<br />

y =−2<br />

<br />

<br />

x = 4 (4, −5). <br />

<br />

<br />

x = 2 (2, −1). <br />

<br />

<br />

x =5 (3, −2). x =5<br />

(3, −2).


(3, −2).<br />

<br />

−2. <br />

x =5 y =−2.<br />

<br />

y =−3 (−3, 5). <br />

<br />

<br />

y =−3 x = a. <br />

(−3, 5), −3. <br />

x =−3<br />

<br />

<br />

y = 1 (−5, 1).


y = −5 (−4, −5).


y (0, 5)<br />

<br />

(0, −6)<br />

−3 <br />

y (0, −1)<br />

−1 <br />

y (0, 3)<br />

1 5 <br />

y (0, −5)<br />

− 3 4 <br />

y (0, −2)<br />

<br />

y (0, −1)<br />

−4 <br />

y (0, 0)


m = 5 8 , (8, 3) m = 5 6 , (6, 7) m =−3 , (10, −5)<br />

5<br />

m =− 3 4 , (8, −5) m =−3 2 , (−4, −3) m =−5 , (−8, −2)<br />

2<br />

m =−7, (−1, −3) m =−4, (−2, −3) <br />

(−2, 5)<br />

<br />

(−2, −3)<br />

<br />

(−1, −7)<br />

<br />

(4, −8)


(2, 6) (5, 3) (4, 3) (8, 1) (−3, −4) (5−2).<br />

(−5, −3) (4, −6). (−1, 3) (−6, −7). (−2, 8) (−4, −6).<br />

(0, 4) (2, −3). (0, −2) (−5, −3). (7, 2) (7, −2).<br />

(−2, 1) (−2, −4). (3, −4) (5, −4). (−6, −3) (−1, −3)<br />

<br />

<br />

<br />

y =4x +2,<br />

y =−3x −1,<br />

2x − y =6,<br />

(1, 2)<br />

(2, −3).<br />

(3, 0).<br />

2x +3y =6,<br />

(0, 5).<br />

x =−4,<br />

(−3, −5).<br />

x −2=0,<br />

(1, −2)<br />

y =5,<br />

(2, −2)<br />

y +2=0,<br />

(3, −3)<br />

<br />

<br />

<br />

y =−2x +3,<br />

y =−x +5,<br />

y = 3 4 x −2,<br />

(2, 2)<br />

(3, 3)<br />

(−3, 4)<br />

y = 2 3 x −4,<br />

(2, −4)<br />

2x −3y =8,<br />

(4, −1)<br />

4x −3y =5,<br />

(−3, 2)<br />

2x +5y =6,<br />

(0, 0)<br />

4x +5y =−3,<br />

(0, 0)<br />

x =3,<br />

(3, 4)<br />

x =−5,<br />

(1, −2)<br />

x =7,<br />

(−3, −4)<br />

x =−1,<br />

(−4, 0)<br />

y −3=0,<br />

(−2, −4)<br />

y −6=0,<br />

(−5, −3)<br />

<br />

(3, 4)<br />

<br />

(2, 1)


(4, 3)<br />

(8, 1)<br />

<br />

(−2, 0) (−3, −2)<br />

m = 1 , <br />

6<br />

(6, 1)<br />

m = 5 , <br />

6<br />

(6, 7)<br />

<br />

4x +3y =6, <br />

(0, −3)<br />

<br />

2x +3y =6, <br />

(0, 5)<br />

m =− 3 4 ,<br />

(8, −5)<br />

<br />

m =− 3 5 ,<br />

(10, −5)<br />

<br />

<br />

y −1=0, (−2, 6)<br />

<br />

(−6, 2)<br />

x =−3,<br />

(−2, −1)<br />

x =−4,<br />

(−3, −5)<br />

<br />

(−3, −4) (2, −5)<br />

<br />

(−5, −3) (4, −6)<br />

<br />

x −2y =5, (−2, 2)<br />

<br />

4x +3y =1, (0, 0)<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


x > 2 <br />

<br />

<br />

4x + 3 > 23 <br />

<br />

<br />

8 < x > 3 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

Ax + By > C Ax+ By ≥ C Ax+ By < C Ax+ By ≤ C<br />

x >3 <br />

<br />

<br />

<br />

(x, y) <br />

<br />

<br />

(x, y)


y > x +4:<br />

a (0, 0) b (1, 6) c (2, 6) d (−5, −15) e (−8, 12)<br />

a<br />

<br />

(0, 0)<br />

<br />

(0, 0) y > x +4.<br />

b<br />

(1, 6)<br />

<br />

(1, 6) y > x +4.<br />

c<br />

(2, 6)<br />

<br />

(2, 6) y > x +4.<br />

d<br />

(−5, −15)<br />

<br />

(−5, −15) y > x +4.<br />

e


(−8, 12)<br />

<br />

(−8, 12) y > x +4.<br />

y > x − 3 :<br />

a (0, 0) b (4, 9) c (−2, 1) d (−5, −3) e (5, 1)<br />

y < x + 1 :<br />

a (0, 0) b (8, 6) c (−2, −1) d (3, 4) e (−1, −4)<br />

<br />

<br />

x =3 <br />

<br />

<br />

x > 3 <br />

x = 3.<br />

y = x +4 y < x +4. <br />

y > x +4. y = x +4 <br />

<br />

Ax + By = C Ax + By > C <br />

Ax + By < C.<br />

<br />

<br />

<br />

<br />

Ax + By < C<br />

Ax+ By ≤ C<br />

Ax + By > C<br />

Ax+ By ≥ C<br />

Boundary line is Ax + By = C<br />

Boundary line is Ax + By = C<br />

Boundary line is not included in solution. Boundary line is included in solution.<br />

Boundary line is dashed.<br />

Boundary line is solid.<br />

y = x +4,


y > x +4 <br />

y > x +4?<br />

(1, 6) (−8, 12) y > x +4. <br />

y = x +4.<br />

(0, 0) (−5, −15) y = x +4,<br />

y > x +4. y < x +4.<br />

<br />

(2, 6)? 6=2+4, y = x +4, <br />

y > x +4. (2, 6) <br />

y > x +4. <br />

(0, 10) <br />

(0, 10) y > x +4.<br />

y > x +4<br />

10 > ? 0+4<br />

10 > 4<br />

y > x +4. <br />

<br />

(0, 0) (−5, −15), y > x +4,


y > x +4 <br />

y = x +4 y > x +4.<br />

y = x +4, y > x +4,


y =2x −1. <br />

<br />

y =2x −1 y >2x −1 <br />

y 2x −1 y 2x −1<br />

y ? 2·0−1 0< ? 2·0−1<br />

0 > −1 True 0 < −1 False<br />

y >2x −1 (0, 0),<br />

y >2x −1.<br />

<br />

<br />

y ≥2x −1.<br />

(0, 0)? <br />

y


y =−2x + 3.<br />

<br />

y = 1 2 x − 4.<br />

<br />

2x +3y =6.


2x +3y =6 2x +3y >6 <br />

2x +3y 6 2x +3y 6<br />

2(0) +3(0) > ? 6<br />

0 > 6 False<br />

(0, 0) 2x +3y 6.<br />

<br />

2x +3y


3x − y = 6.<br />

<br />

<br />

<br />

<br />

<br />

y ≥ 3 4 x −2.


y ≥ 5 2 x − 4.<br />

<br />

y ≤ 2 3 x − 5.<br />

<br />

<br />

<br />

<br />

≤ or ≥ , <br />

< or > , <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

x −2y


(0, 0) <br />

(0, 0) x −2y


y ≤−4x.<br />

<br />

y =−4x. m =−4 b =0. <br />

<br />

≤<br />

(1, 0) <br />

(1, 0) y ≤−4x?<br />

(1, 0) y ≤−4x, <br />

y ≤−4x.<br />

y > − 3x.<br />

<br />

y ≥ −2x.


x = a<br />

y = b<br />

vertical line<br />

horizontal line<br />

<br />

y >3.<br />

<br />

y =3. <br />

(0, 0).<br />

> <br />

(0, 0) y >3.<br />

y >3<br />

0>3<br />

(0, 0) <br />

y >3.<br />

y < 5.<br />

y ≤ −1.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a x


c (x, y) <br />

<br />

a <br />

<br />

<br />

<br />

b <br />

10x +15y ≥ 240<br />

15y ≥ −10x + 240<br />

y ≥ − 2 3 x +16<br />

c (15, 10), (0, 16), (24, 0)


a <br />

<br />

b <br />

c <br />

<br />

<br />

<br />

<br />

<br />

a <br />

<br />

b <br />

c


y > x −1:<br />

<br />

<br />

y > x −3:<br />

<br />

<br />

y 2:<br />

a (−3, 0)<br />

b (1, 6)<br />

c (−6, −2)<br />

d (0, 1)<br />

e (5, −4)<br />

a (5, 1)<br />

b (−2, 6)<br />

c (3, 2)<br />

d (10, −5)<br />

e (0, 0)<br />

a (1, 1)<br />

b (4, −3)<br />

c (0, 0)<br />

d (−8, 12)<br />

e (3, 0)<br />

<br />

<br />

<br />

<br />

y =3x −4.<br />

<br />

<br />

y =2x −4.<br />

<br />

<br />

y = 1 2 x +1.


y =− 1 3 x −2.<br />

<br />

<br />

<br />

x + y =5.<br />

<br />

<br />

<br />

x + y =3.<br />

<br />

<br />

<br />

3x − y =6.<br />

<br />

<br />

<br />

2x − y =4.<br />

<br />

<br />

<br />

y > 2 3 x −1.<br />

<br />

y < 3 5 x +2.<br />

<br />

y ≤− 1 2 x +4.<br />

<br />

y ≥− 1 3 x −2.<br />

<br />

x − y ≤3.<br />

<br />

x − y ≥−2.<br />

<br />

4x + y >−4.<br />

<br />

x +5y 4x.<br />

<br />

y ≤−3x.<br />

<br />

y < − 10.<br />

<br />

y ≥2.<br />

<br />

x ≤5.<br />

<br />

x ≥0.<br />

<br />

x − y


y ≥ 3 2 x.<br />

<br />

y ≤ 5 4 x.<br />

<br />

y >−2x +1.<br />

<br />

y 10.<br />

<br />

4x −3y > 12.<br />

<br />

<br />

<br />

$16.50 <br />

<br />

<br />

<br />

a <br />

<br />

<br />

<br />

b <br />

c (x, y) <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

<br />

<br />

<br />

b <br />

c (x, y) <br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

<br />

<br />

b <br />

c <br />

<br />

<br />

<br />

<br />

<br />

a <br />

<br />

<br />

<br />

b <br />

c <br />

<br />

<br />

<br />

> <br />


a <br />

b


3x − 5 x = −2 <br />

<br />

2x 2 − x − 3 x = a.<br />

<br />

7x − 1 − 4x + 5.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

(Student name, Student ID #)<br />

<br />

<br />

<br />

<br />

<br />

(x, y). <br />

<br />

<br />

{(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)} :<br />

a <br />

b <br />

<br />

{(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)}<br />

a {1, 2, 3, 4, 5}


{1, 4, 9, 16, 25}<br />

{(1, 1), (2, 8), (3, 27), (4, 64), (5, 125)} :<br />

a <br />

b <br />

{(1, 3), (2, 6), (3, 9), (4, 12), (5, 15)} :<br />

a <br />

b <br />

<br />

<br />

<br />

<br />

a b <br />

c <br />

<br />

a <br />

<br />

<br />

<br />

b <br />

<br />

c


a b <br />

c <br />

<br />

a b <br />

c <br />

<br />

<br />

<br />

<br />

a b c <br />

<br />

<br />

a {(1, 5), (−3, −1), (4, −2), (0, 3), (2, −2), (−3, 4)}.<br />

b <br />

{−3,0,1,2,4}.<br />

−3 <br />

c {−2, −1, 3, 4, 5}.<br />

−2


a b c <br />

<br />

<br />

a b c <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

(−3, −1) (−3, 4). <br />

<br />

<br />

<br />

<br />

a {(−3, 27), (−2, 8), (−1, 1), (0, 0), (1, 1), (2, 8), (3, 27)}


{(9, −3), (4, −2), (1, −1), (0, 0), (1, 1), (4, 2), (9, 3)}<br />

<br />

a {(−3, 27), (−2, 8), (−1, 1), (0, 0), (1, 1), (2, 8), (3, 27)}<br />

<br />

<br />

{−3,−2,−1,0,1,2,3}.<br />

<br />

{27, 8, 1, 0}.<br />

b {(9, −3), (4, −2), (1, −1), (0, 0), (1, 1), (4, 2), (9, 3)}<br />

−3. <br />

<br />

{0, 1, 2, 4, 9}.<br />

<br />

{−3,−2,−1,0,1,2,3}.<br />

<br />

<br />

<br />

a {(−3, −6), (−2, −4), (−1, −2), (0, 0), (1, 2), (2, 4), (3, 6)}<br />

b {(8, −4), (4, −2), (2, −1), (0, 0), (2, 1), (4, 2), (8, 4)}<br />

<br />

<br />

<br />

a {(27, −3), (8, −2), (1, −1), (0, 0), (1, 1), (8, 2), (27, 3)}<br />

b {(7, −3), (−5, −4), (8, −0), (0, 0), (−6, 4), (−2, 2), (−1, 3)}<br />

<br />

a b c <br />

<br />

<br />

a <br />

<br />

b


c <br />

{321-549-3327, 427-658-2314, 321-964-7324, 684-358-7961, 684-369-7231, 798-367-8541}<br />

<br />

a b c <br />

<br />

<br />

a b c <br />

<br />

<br />

<br />

<br />

<br />

a 2x + y =7b y = x 2 +1 c x + y 2 =3<br />

<br />

a 2x + y =7<br />

−2 <br />

x =3:


x =3, y =1. <br />

<br />

b y = x 2 +1<br />

<br />

x =2:<br />

x =2, y =5. <br />

<br />

c<br />

<br />

x =2.<br />

y =1y =−1<br />

x =2, y =1 y =−1. <br />

<br />

<br />

<br />

a 4x + y = −3 b x + y 2 = 1 c y − x 2 = 2<br />

<br />

<br />

a x + y 2 = 4 b y = x 2 − 7 c y = 5x − 4<br />

<br />

<br />

f (x). <br />

<br />

<br />

y = f (x)<br />

f (x) <br />

f is the name of the function<br />

x is the domain value<br />

f (x) is the range value y corresponding to the value x


y = f (x),<br />

x is the independent variable as it can be any value in the domain<br />

y the dependent variable as its value depends on x<br />

<br />

<br />

y =4x −5. x =2, x =2 <br />

<br />

x =2.<br />

x =2 <br />

y =4x −5 <br />

x =2, <br />

f (x) =4x −5. <br />

x =2.<br />

x =2 <br />

f (x) <br />

<br />

f (x) =2x 2 +3x −1, <br />

a f (3) b f (−2) c f (a)<br />

a<br />

<br />

f (3),<br />

<br />

<br />

b


c<br />

f (a),<br />

<br />

<br />

<br />

f (x) = 3x 2 − 2x + 1, <br />

a f (3) b f (−1) c f (t)<br />

<br />

f (x) = 2x 2 + 4x − 3, <br />

a f (2) b f (−3) c f (h)<br />

f (x) g(x) <br />

<br />

<br />

g(x) =3x −5, <br />

a g(h 2 ) b g(x +2) c g(x) + g(2)<br />

a<br />

<br />

g ⎛ ⎝h 2⎞ ⎠,<br />

h 2 <br />

b


g(x +2),<br />

x +2 <br />

<br />

c<br />

g(x) + g(2),<br />

g(2).<br />

<br />

b c g(x +2)=3x +1 g(x) + g(2) =3x −4. <br />

g(x +2)≠g(x) + g(2).<br />

<br />

g(x) = 4x − 7, <br />

a g(m 2 ) b g(x − 3) c g(x) − g(3)<br />

<br />

h(x) = 2x + 1, <br />

a h(k 2 ) b h(x + 1) c h(x) + h(1)<br />

<br />

<br />

<br />

N(t) =75+10t <br />

a <br />

b N(5). <br />

<br />

a <br />

t <br />

b N(5).


t =5.<br />

<br />

N(5),<br />

<br />

<br />

<br />

<br />

N(t) = 100 + 15t <br />

<br />

<br />

a <br />

b N(7). <br />

<br />

<br />

N(t) = 110 +25t <br />

<br />

<br />

a <br />

b N(14).


a b <br />

{(1, 4), (2, 8), (3, 12), (4, 16), (5, 20) ⎫ ⎭ ⎬<br />

{(1, −2), (2, −4), (3, −6), (4, −8), ⎛ ⎝5, −10)}<br />

{(1, 7), (5, 3), (7, 9), (−2, −3), (−2, 8)} {(11, 3), (−2, −7), (4, −8), (4, 17), (−6, 9)}<br />

a b <br />

c <br />

<br />

<br />

5′4 ″ <br />

<br />

<br />

18.5 – 24.9 <br />

<br />

5′11′′ <br />

<br />

<br />

18.5 – 24.9


a b <br />

c <br />

<br />

<br />

<br />

<br />

<br />

a b <br />

c <br />

{(−3, 9), (−2, 4), (−1, 1),<br />

(0, 0), (1, 1), (2, 4), (3, 9)}<br />

{(9, −3), (4, −2), (1, −1),<br />

(0, 0), (1, 1), (4, 2), (9, 3)}<br />

{(−3, 27), (−2, 8), (−1, 1),<br />

(0, 0), (1, 1), (2, 8), (3, 27)}<br />

{(−3, −27), (−2, −8), (−1, −1),<br />

(0, 0), (1, 1), (2, 8), (3, 27)}


a b <br />

c <br />

<br />

<br />

<br />

<br />

<br />

<br />

a 2x + y =−3<br />

b y = x 2<br />

c x + y 2 =−5<br />

<br />

a y =3x −5<br />

b y = x 3<br />

c 2x + y 2 =4<br />

<br />

a y −3x 3 =2<br />

b x + y 2 =3<br />

c 3x −2y =6<br />

<br />

a 2x −4y =8<br />

b −4 = x 2 − y<br />

c y 2 =−x +5<br />

<br />

a f(2) b f (−1) c f (a).<br />

f (x) =5x −3 f (x) =3x +4 f (x) =−4x +2<br />

f (x) =−6x −3 f (x) =x 2 − x +3 f (x) =x 2 + x −2<br />

f (x) =2x 2 − x +3<br />

f (x) =3x 2 + x −2<br />

a g(h 2 ) b g(x + 2) c g(x) +g(2).<br />

g(x) =2x +1 g(x) =5x −8 g(x) =−3x −2<br />

g(x) =−8x +2 g(x) =3−x g(x) =7−5x


f (x) =3x 2 −5x; f (2) g(x) =4x 2 −3x; g(3) F(x) =2x 2 −3x +1;<br />

F(−1)<br />

G(x) =3x 2 −5x +2;<br />

G(−2)<br />

h(t) =2|t −5| +4; h(−4)<br />

h(y) =3|y −1| −3; h(−4)<br />

f (x) =<br />

x x +2 ; f (2) g(x) =x −2<br />

−1 x +2 ; g(4)<br />

<br />

<br />

<br />

N(t) =85+20t <br />

<br />

<br />

a <br />

<br />

b N(4). <br />

<br />

<br />

N(t) =43+t <br />

<br />

<br />

a <br />

<br />

b N(30). <br />

<br />

<br />

C(x) = 3.25x + 1500 <br />

<br />

a <br />

<br />

b C(0). <br />

c C(1000). <br />

<br />

C(x) = 7.25x + 2500 <br />

C(x) <br />

<br />

a <br />

<br />

b C(0). <br />

c C(1000). <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


a 2 3 b 3 2 .<br />

<br />

<br />

a |7| b |−3|.<br />

<br />

<br />

a 4 b 16.<br />

<br />

<br />

<br />

<br />

(x, y) <br />

<br />

<br />

<br />

y =2x −3,


y =2x −3 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


y =2x −3 <br />

y =2x −3 <br />

f (x) =2x −3. <br />

(x, y) y = f (x). ⎛ ⎝x, f (x) ⎞ ⎠.


y =2x −3 <br />

<br />

f (x) =2x −3 <br />

<br />

<br />

(x, y) <br />

y = f (x).<br />

f name of function<br />

x x-coordinate of the ordered pair<br />

f (x) y-coordinate of the ordered pair<br />

⎛<br />

⎝x, f (x) ⎞ ⎠<br />

<br />

<br />

<br />

<br />

y =2x −3 <br />

<br />

<br />

y = mx + b. <br />

f (x) = mx + b


f (x) =−2x −4.<br />

<br />

<br />

f (x) =−2x −4<br />

m =−2<br />

b =−4<br />

<br />

f (x) = −3x − 1<br />

f (x) = −4x − 5<br />

f (x) = b,


f (x) y = b. <br />

f (x) = b, <br />

<br />

<br />

<br />

<br />

f (x) =4.<br />

<br />

f (x) =4<br />

<br />

(0, 4).<br />

f (x) = −2.<br />

f (x) = 3.<br />

<br />

f (x) = x <br />

f (x) =1x +0,


f (x) = x 2 .<br />

<br />

<br />

<br />

f (x) = x 2 .<br />

f (x) = −x 2


f (x) = x 3 .<br />

<br />

<br />

<br />

f (x) = x 3 .<br />

<br />

f (x) = −x 3 .


f (x) = x <br />

<br />

<br />

f (x) =<br />

x<br />

<br />

<br />

<br />

f (x) = x.<br />

f (x) = − x.


f (x) = |x|. <br />

<br />

<br />

<br />

f (x) = |x|.<br />

<br />

<br />

<br />

f (x) = |x|.<br />

<br />

f (x) = −|x|.


[−3, 3].<br />

<br />

[−1, 3].


a f (0).<br />

b f ⎛ ⎝ 3 2 π⎞ ⎠ .<br />

c f ⎛ ⎝ −1 2 π⎞ ⎠ .<br />

d f (x) =0.<br />

e <br />

f <br />

g <br />

h <br />

<br />

a x =0, f (0) =0.


x = 3 2 π, −1. f ⎛ ⎝ 3 2 π⎞ ⎠ =−1.<br />

c x =− 1 2 π, −1. f ⎛ ⎝ −1 2 π⎞ ⎠ =−1.<br />

d (−2π, 0), (−π, 0), (0, 0), (π, 0), (2π, 0). f (x) =0 <br />

−2π, −π, 0,π, 2π.<br />

e y =0. f (x) =0. <br />

(−2π, 0), (−π, 0), (0, 0), (π, 0), (2π, 0).<br />

f x =0. f (0). (0, 0).<br />

g (−∞, ∞)<br />

h −1 [−1, 1].<br />

<br />

<br />

a f (0).<br />

b f ⎛1 ⎞ ⎝ π 2 ⎠.<br />

c f ⎛ ⎝ ⎛ − 3<br />

2<br />

π ⎞ ⎠ ⎞ .<br />

d f (x) = 0.<br />

e <br />

f <br />

g <br />

h


a f (0).<br />

b f (π).<br />

c f (−π).<br />

d f (x) = 0.<br />

e <br />

f <br />

g <br />

h


a<br />

a<br />

b<br />

b<br />

a<br />

a<br />

b<br />

b


a b <br />

f (x) =3x +4 f (x) =2x +5 f (x) =−x −2<br />

f (x) =−4x −3 f (x) =−2x +2 f (x) =−3x +3<br />

f (x) = 1 2 x +1 f (x) = 2 3 x −2 f (x) =5<br />

f (x) =2 f (x) =−3 f (x) =−1<br />

f (x) =2x f (x) =3x f (x) =−2x<br />

f (x) =−3x f (x) =3x 2 f (x) =2x 2<br />

f (x) =−3x 2 f (x) =−2x 2 f (x) = 1 2 x2<br />

f (x) = 1 3 x2 f (x) = x 2 −1 f (x) = x 2 +1<br />

f (x) =−2x 3 f (x) =2x 3 f (x) = x 3 +2<br />

f (x) = x 3 −2 f (x) =2 x f (x) =−2 x<br />

f (x) = x −1 f (x) = x +1 f (x) =3|x|<br />

f (x) =−2|x| f (x) = |x| +1 f (x) = |x| −1


a f (0).<br />

b f ⎛ ⎝ 1 2 π⎞ ⎠ .<br />

c f ⎛ ⎝ −3 2 π⎞ ⎠ .<br />

d f (x) =0.<br />

e <br />

f <br />

g <br />

h <br />

a f (0).<br />

b f (π).<br />

c f (−π).<br />

d f (x) =0.<br />

e <br />

f <br />

g <br />

h


a f (0).<br />

b f (−3).<br />

c f (3).<br />

d f (x) =0.<br />

e <br />

f <br />

g <br />

h <br />

a f (0).<br />

b f (x) =0.<br />

c <br />

d <br />

e <br />

f <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


Ax + By = C <br />

Ax + By > C Ax + By < C.<br />

<br />

<br />

y = b. <br />

(0, b).<br />

<br />

Ax + By = C, <br />

<br />

<br />

Ax + By > C, Ax + By ≥ C, Ax + By < C, Ax + By ≤ C, <br />

<br />

<br />

(x, y) <br />

<br />

(0, 0) <br />

<br />

<br />

(x 1 , y 1 ) <br />

y − y 1 = m(x − x 1 ).<br />

<br />

(x, y). <br />

<br />

(x, y) <br />

Ax + By = C, <br />

<br />

(x, y) <br />

<br />

Ax + By = C.<br />

<br />

x = a. (a, 0).<br />

<br />

<br />

<br />

<br />

(a, 0).<br />

0 (0, b).<br />

<br />

<br />

Quadrant I Quadrant II Quadrant III Quadrant IV<br />

⎛<br />

⎝ x, y ⎞ ⎠<br />

⎛<br />

⎝ x, y ⎞ ⎠<br />

⎛<br />

⎝ x, y ⎞ ⎠<br />

⎛<br />

⎝ x, y ⎞ ⎠<br />

⎛<br />

⎝ +, + ⎞ ⎠<br />

⎛<br />

⎝ −, + ⎞ ⎠<br />

⎛<br />

⎝ −, − ⎞ ⎠<br />

⎛<br />

⎝ +, − ⎞ ⎠


Ax + By = C <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

(a, 0) <br />

<br />

(0, b) <br />

<br />

<br />

<br />

<br />

y =0 <br />

x =0 <br />

<br />

<br />

y =0 <br />

x =0 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

m = rise run .


m = rise run .<br />

<br />

<br />

<br />

<br />

m = rise run .<br />

<br />

<br />

<br />

<br />

(x 1 , y 1 ) (x 2 , y 2 ) <br />

m = y 2 − y 1<br />

x 2 − x 1<br />

.<br />

<br />

<br />

<br />

m = rise run <br />

<br />

<br />

<br />

(0, b) y = mx + b<br />

<br />

<br />

<br />

<br />

<br />

m 1 m 2 m 1 = m 2 .<br />

<br />

<br />

<br />

m 1 m 2 <br />

m 1 =−m 1 . 2<br />

−1, m 1 · m 2 =−1.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

y − y 1 = m(x − x 1 ).


m = y 2 − y 1<br />

x 2 − x 1<br />

<br />

y − y 1 = m(x − x 1 ).<br />

<br />

<br />

<br />

y = mx + b<br />

y − y 1 = m(x − x 1 )<br />

y − y 1 = m(x − x 1 )<br />

<br />

<br />

<br />

<br />

<br />

<br />

y − y 1 = m(x − x 1 ).<br />

<br />

<br />

<br />

<br />

<br />

y − y 1 = m(x − x 1 )<br />

<br />

<br />

<br />

<br />

<br />

≤or≥, <br />

, <br />

<br />

<br />

<br />

<br />

<br />

<br />

y = f (x)<br />

<br />

<br />

<br />

<br />

<br />

f (x)


f (x) <br />

<br />

y = f (x),<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

(x, y) <br />

y = f (x).<br />

<br />

f<br />

x<br />

f (x)<br />

name of function<br />

x-coordinate of the ordered pair<br />

y-coordinate of the ordered pair<br />

⎛<br />

⎝x, f (x) ⎞ ⎠


a (−1, −5)<br />

b (−3, 4)<br />

c (2, −3)<br />

d ⎛ ⎝ 1, 5 ⎞<br />

2⎠<br />

<br />

a (−2, 0)<br />

b (0, −4)<br />

c (0, 5)<br />

d (3, 0)<br />

<br />

5x + y = 10;<br />

y =6x −2;<br />

a (5, 1)<br />

b (2, 0)<br />

c (4, −10)<br />

a (1, 4)<br />

b ⎛ ⎝ 1 3 ,0⎞ ⎠<br />

c (6, −2)<br />

<br />

<br />

y =4x −3 y =−3x y = 1 2 x +3<br />

y =− 4 5 x −1 x − y =6 2x + y =7<br />

3x −2y =6<br />

<br />

<br />

y =−2 x =3<br />

<br />

y =−2x y =−2 y = 4 3 x y = 4 3


x − y =−1 x +2y =6 2x +3y =12<br />

y = 3 4 x −12 y =3x<br />

<br />

<br />

−x +3y =3 x − y =4 2x − y =5<br />

2x −4y =8 y =4x


y =2 x =5 x =−3<br />

y =−1<br />

<br />

<br />

(−1, −1), (0, 5) (3,5), (4, −1) (−5, −2), (3, 2)<br />

(2, 1), (4, 6)


(2, −2); m = 5 (−3, 4); m =−<br />

2<br />

1 3<br />

−4; m =3<br />

m =− 3 4<br />

<br />

<br />

y =−4x +9 y = 5 3 x −6 5x + y =10<br />

4x −5y =8<br />

<br />

y =2x +3 y =−x −1 y =− 2 5 x +3<br />

4x −3y =12<br />

<br />

x =5 y =−3 2x + y =5<br />

x − y =2 y = 2 2 x +2 y = 3 4 x −1<br />

<br />

<br />

C = 6.5m +42 <br />

<br />

<br />

a <br />

<br />

b <br />

c <br />

d <br />

<br />

P =35h − 250 <br />

<br />

<br />

a <br />

<br />

b <br />

<br />

c <br />

d <br />

<br />

<br />

4x −3y =−1;y = 4 3 x −3 y =5x −1;10x +2y =0 3x −2y =5;2x +3y =6<br />

2x − y =8;x −2y =4


1 y <br />

−5 y y <br />

3<br />

(0, −3)<br />

(0, 4)<br />

(0, −6)<br />

−2 y <br />

(0, 0)<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

m =− 1 4 , (−8, 3) m = 3 , (10, 6)<br />

5<br />

(−2, 7) m =−2, (−1, −3)


(2, 10) (−2, −2) (7, 1) (5, 0)<br />

(3, 8) (3, −4) (5, 2) (−1, 2)<br />

<br />

<br />

<br />

y =−3x +6, (1, −5) 2x +5y = −10, (10, 4)<br />

x =4, (−2, −1) y =−5, (−4, 3)<br />

<br />

<br />

<br />

y =− 4 5 x +2, (8, 9) 2x −3y =9, (−4, 0)<br />

y =3, (−1, −3) x =−5 (2, 1)<br />

<br />

<br />

<br />

<br />

y < x −3:<br />

<br />

x + y >4:<br />

a (0, 1) b (−2, −4) c (5, 2) d (3, −1)<br />

e (−1, −5)<br />

a (6, 1) b (−3, 6) c (3, 2) d (−5, 10) e (0, 0)<br />

<br />

<br />

<br />

y =−x +2.<br />

y = 2 3 x −3.


x + y =−4.<br />

<br />

x −2y =6.<br />

<br />

<br />

<br />

y > 2 5 x −4. y ≤− 1 4 x +3. x − y ≤5.<br />

<br />

3x +2y > 10.<br />

<br />

y ≤−3x.<br />

<br />

y


a b <br />

c <br />

<br />

<br />

<br />

a b <br />

c <br />

<br />

<br />

a b <br />

c <br />

{(9, −5), (4, −3), (1, −1),<br />

(0, 0), (1, 1), (4, 3), (9, 5)}<br />

{(−3, 27), (−2, 8), (−1, 1),<br />

(0, 0), (1, 1), (2, 8), (3, 27)}<br />

a b <br />

c


2x + y =−3 y = x 2 y =3x −5<br />

y = x 3 2x + y 2 =4<br />

<br />

<br />

a f (−2) b f(3) c f (a).<br />

f (x) =3x −4 f (x) =−2x +5 f (x) =x 2 −5x +6<br />

f (x) =3x 2 −2x +1<br />

<br />

g(x) =3x 2 −5x; g(2) F(x) =2x 2 −3x +1;<br />

F(−1)<br />

<br />

h(t) =4|t −1| +2; h(−3)<br />

f (x) = x +2<br />

x −1 ; f (3)


a b <br />

f (x) =5x +1 f (x) =−4x −2 f (x) = 2 3 x −1<br />

f (x) =−6 f (x) =2x f (x) =3x 2<br />

f (x) =− 1 2 x2 f (x) = x 2 +2 f (x) = x 3 −2<br />

f (x) = x +2 f (x) =−|x| f (x) = |x| +1


a f (0).<br />

b f ⎛ ⎝ 1 2 π⎞ ⎠ .<br />

c f ⎛ ⎝ −3 2 π⎞ ⎠ .<br />

d <br />

f (x) =0.<br />

e x <br />

f y <br />

g <br />

<br />

h <br />

<br />

a f (0).<br />

b <br />

f (x) =0.<br />

c x <br />

d y <br />

e <br />

<br />

f


a (2, 5)<br />

b (−1, −3)<br />

c (0, 2)<br />

d ⎛ ⎝ −4, 3 ⎞<br />

2⎠<br />

e (5, 0)<br />

<br />

<br />

3x − y =6?<br />

a (3, 3) b (2, 0) c (4, −6)<br />

<br />

a<br />

<br />

(5, 2) <br />

(−1, −4).<br />

1 2<br />

(−3, −4).<br />

b<br />

<br />

4x +2y =−8


y = 5 3 x −1 y =−x y =2<br />

<br />

− 3 y <br />

m =2, (−3, −1) (10, 1) <br />

4<br />

(6, −1)<br />

(0, −2)<br />

<br />

y = 5 4 x +2,<br />

(−10, 3)<br />

<br />

<br />

<br />

y =−x −3.<br />

<br />

y > 3 2 x +5 x − y ≥−4<br />

y ≤−5x <br />

<br />

<br />

<br />

<br />

<br />

a <br />

<br />

<br />

<br />

b <br />

c (x, y) <br />

<br />

<br />

<br />

a <br />

b <br />

c <br />

{(−3, 27), (−2, 8), (−1, 1), (0, 0),<br />

(1, 1), (2, 8), (3, 27)}<br />

<br />

f (c).<br />

a f (−1) b f (2) c<br />

f (x) =4x 2 −2x −3


h(y) =3|y −1| −3, h(−4). <br />

<br />

a b <br />

<br />

f (x) = x 2 +1 f (x) = x +1<br />

<br />

b y <br />

c f (−1).<br />

d f (1).<br />

e <br />

<br />

f


y = 2 3 x − 4,<br />

a (6, 0) b (−3, −2)


3x − y = 12.<br />

<br />

2x − 3y = 12.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧2x + y =7<br />

⎨<br />

⎩x −2y =6<br />

2x + y =7, <br />

<br />

<br />

(x, y) <br />

<br />

<br />

<br />

(x, y).<br />

<br />

<br />

<br />

<br />

⎧ x − y =−1<br />

⎨<br />

⎩2x − y =−5 .<br />

a (−2, −1) b (−4, −3)<br />

a


<br />

<br />

⎧3x + y = 0<br />

⎨<br />

⎩ x + 2y = −5 .<br />

a (1, −3) b (0, 0)<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x − 3y = −8<br />

−3x − y = 4 .<br />

a (2, −2) b (−2, 2)<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧2x + y =7<br />

⎨<br />

⎩x −2y =6 .


⎧x<br />

− 3y = −3<br />

⎨ .<br />

⎩ x + y = 5<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−x + y = 1<br />

3x + 2y =12 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x + y =−1<br />

.<br />

2x + y =0<br />

<br />

y


(−1, 2).<br />

<br />

(−1, 2).<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−x + y = 1<br />

2x + y = 10 .<br />

<br />

<br />

⎧2x + y = 6<br />

⎨<br />

⎩ x + y = 1 .<br />

<br />

<br />

<br />

<br />

<br />

⎧<br />

⎨ y = 1 2 x −3 .<br />

⎩x −2y =4


⎧ y = − 1x + 2<br />

⎨ 4 .<br />

⎩ x + 4y = − 8<br />

<br />

<br />

⎧ y = 3x − 1<br />

⎨<br />

⎩6x − 2y = 6 .<br />

<br />

<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

y =2x −3<br />

−6x +3y =−9 .


⎧ y = − 3x − 6<br />

⎨<br />

⎩6x + 2y = − 12 .<br />

<br />

<br />

⎧ y = 1x − 4<br />

⎨ 2 .<br />

⎩2x − 4y = 16


a ⎧ =3x −1<br />

⎨y<br />

⎩6x −2y =12 b ⎧ + y =−3<br />

⎨2x<br />

⎩x −5y =5<br />

<br />

a <br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

y = 3x −1<br />

6x −2y = 12<br />

y = 3x −1<br />

6x −2y = 12<br />

−2y = −6x +12<br />

−2y<br />

= −6x +12<br />

−2 −2<br />

y = 3x −6<br />

y = 3x −1 y = 3x −6<br />

m = 3 m = 3<br />

b = −1 b = −6<br />

<br />

<br />

<br />

b


⎧2x + y = −3<br />

⎨<br />

⎩x −5y = 5<br />

<br />

2x + y = −3 x −5y = 5<br />

y = −2x −3 −5y = −x +5<br />

−5y<br />

−5<br />

=<br />

−x +5<br />

−5<br />

y = 1 5 x −1<br />

y = −2x −3 y = 1 5 x −1<br />

<br />

m = −2 m = 1 5<br />

b = −3 b = −1<br />

<br />

<br />

<br />

<br />

a ⎨⎧<br />

⎧<br />

⎩<br />

y = −2x −4<br />

4x + 2y = 9 b ⎧ 3x + 2y = 2<br />

⎨⎧<br />

⎩2x + y = 1<br />

<br />

<br />

a ⎨⎧<br />

⎧ y = 1x − 5 3 ⎩x<br />

− 3y = 6<br />

b ⎧ ⎨⎧<br />

⎩<br />

⎧ x + 4y =12<br />

−x + y = 3<br />

<br />

<br />

−10 <br />

<br />

<br />

<br />

<br />

⎧2x + y =7<br />

⎨<br />

⎩x −2y =6


⎧2x + y =7<br />

⎨<br />

⎩x −2y =6 .<br />

<br />

<br />

<br />

⎧−2x + y = −11<br />

⎨<br />

.<br />

⎩ x + 3y = 9<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x + y = −1<br />

4x + 3y = 3 .


⎧<br />

⎨<br />

⎩<br />

4x +2y =4<br />

6x − y =8 .<br />

<br />

<br />

<br />

−2x +2 <br />

−2x +2.<br />

<br />

x = 5 4 4x +2y =4 ⎛ ⎝ 5 4 , − 1 ⎞<br />

2⎠ .


⎛ ⎝ 5 4 , − 1 ⎞<br />

2⎠ .<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x − 4y = −4<br />

−3x + 4y = 0 .<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x − y = 0<br />

2x − 3y = 5 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

if a = b<br />

and c = d<br />

then a + c = b + d.<br />

<br />

<br />

<br />

<br />

⎧ 3x + y =5<br />

⎨<br />

⎩—————<br />

2x − y =0<br />

5x =5<br />

<br />

<br />

⎧ x +4y =2<br />

⎨<br />

⎩2x +5y =−2<br />

<br />

−2, <br />

−2.


⎧2x + y =7<br />

⎨<br />

⎩x −2y =6 .<br />

<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x + y = 5<br />

2x − 3y = 7 .<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x + y = − 5<br />

−2x − 2y = − 2 .


⎧<br />

⎨<br />

⎩<br />

4x −3y =9<br />

7x +2y =−6 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

x =0 <br />

<br />

<br />

(0, −3).


(0, −3).<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x − 4y = − 9<br />

5x + 3y = 14 .<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

7x + 8y = 4<br />

3x − 5y = 27 .<br />

<br />

<br />

<br />

⎨ x + 1 2 y =6<br />

3<br />

⎩2 x + 2 3 y = 17 .<br />

2<br />

<br />

⎧<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

y −4.<br />

<br />

x =3 <br />

y <br />

(3, 6).


(3, 6).<br />

<br />

<br />

⎧1<br />

x −<br />

3<br />

1 2<br />

⎨<br />

y = 1<br />

3<br />

⎩ x − y =<br />

4<br />

5 .<br />

2<br />

<br />

<br />

⎧ x + 3y = − 1<br />

⎨ 5 5<br />

⎩− 1 2 x − 2 3 y = 5 .<br />

6<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧3x +4y =12<br />

⎨<br />

⎩y =3− 3 4 x .


⎧3x +4y =12<br />

⎨<br />

⎩y =3− 3 4 x<br />

<br />

<br />

<br />

<br />

⎧3x +4y =12<br />

⎨3<br />

⎩4 x + y =3<br />

⎧3x +4y =12<br />

⎨<br />

⎩4 ⎛ ⎝ 3 4 x + y⎞ ⎠ =4(3)<br />

⎧<br />

⎨<br />

⎩<br />

⎧<br />

⎨<br />

⎩<br />

3x +4y =12<br />

3x +4y =12<br />

3x +4y =12<br />

______________<br />

−3x −4y =−12<br />

0=0<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧5x − 3y = 15<br />

⎨<br />

⎩ y = − 5 + 5 3 x.<br />

<br />

<br />

⎧ x + 2y = 6<br />

⎨<br />

⎩ y = − 1 2 x + 3.<br />

<br />

<br />

<br />

<br />

Choose the Most Convenient Method to Solve a System of Linear Equations<br />

Graphing<br />

————<br />

Substitution<br />

————— Elimination<br />

—————<br />

Use when you need a Use when one equation is Use when the equations are<br />

picture of the situation. already solved or can be in standard form.<br />

easily solved for one<br />

variable.<br />

<br />

<br />

<br />

a ⎧ +8y =40<br />

⎨3x<br />

⎩7x −4y = −32 b ⎧ +6y =12<br />

⎨5x<br />

⎩y = 2 3 x −1


a<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x +8y =40<br />

7x −4y = −32<br />

<br />

b<br />

⎧5x +6y =12<br />

⎨<br />

⎩y = 2 3 x −1<br />

<br />

<br />

<br />

<br />

a ⎧ 4x − 5y = −32<br />

⎨⎧<br />

⎩3x + 2y = −1<br />

b ⎨⎧<br />

⎧ x = 2y − 1<br />

⎩3x − 5y = −7<br />

<br />

<br />

<br />

a ⎨⎧<br />

⎧ y =2x − 1<br />

⎩3x − 4y = − 6 b ⎧ 6x − 2y = 12<br />

⎨⎧<br />

⎩3x + 7y = −13


⎧2x −6y =0<br />

⎧−3x + y =8<br />

⎧x + y =2<br />

⎨ ⎨<br />

⎩3x −4y =5<br />

⎩−x +2y =−9<br />

⎨<br />

⎩y = 3 4 x<br />

a (3, 1) b (−3, 4)<br />

a (−5, −7) b (−5, 7)<br />

a ⎛8 ⎝7 , 6 ⎞<br />

7⎠ b ⎛ ⎝ 1, 3 ⎞<br />

4⎠<br />

⎧2x +3y =6<br />

⎨<br />

⎩y = 2 3 x +2<br />

a (−6, 2) b (−3, 4)<br />

<br />

<br />

⎧3x + y =−3<br />

⎧−x + y =2<br />

⎨ ⎨ <br />

⎩2x +3y =5<br />

⎩2x + y =−4<br />

⎧ y = x +2<br />

⎨<br />

⎩y =−2x +2<br />

<br />

⎧ y = x −2<br />

⎨<br />

⎩y =−3x +2<br />

<br />

⎧<br />

⎨ y = 3 2 x +1<br />

y =− 1 2 x +5<br />

⎩<br />

<br />

⎧<br />

⎨ y = 2 3 x −2<br />

y =− 1 3 x −5<br />

⎩<br />

<br />

⎧ x + y =−4<br />

⎨<br />

⎩−x +2y =−2<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−x +3y =3<br />

x +3y =3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−2x +3y =3<br />

x +3y =12<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x − y =4<br />

2x +3y =12<br />

<br />

⎧x +3y =−6<br />

⎨<br />

⎩y =− 4 3 x +4<br />

<br />

⎧−x +2y =−6<br />

⎨<br />

⎩y =− 1 2 x −1<br />

<br />

⎧−2x +4y =4<br />

⎨<br />

⎩y = 1 2 x<br />

<br />

⎧3x +5y =10<br />

⎨<br />

⎩y =− 3 5 x +1<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x −3y =8<br />

8x −6y =14<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x +3y =4<br />

−2x −6y =3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x =−3y +4<br />

2x +6y =8<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x =3y +7<br />

8x −6y =14<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x + y =6<br />

−8x −4y = −24<br />

<br />

⎧<br />

⎨<br />

⎩<br />

5x +2y =7<br />

−10x −4y = −14<br />

<br />

⎧<br />

⎨ y = 2 3 x +1<br />

⎧<br />

⎨ y = 3 2 x +1<br />

⎧5x +3y =4<br />

⎨<br />

⎩2x −3y =5<br />

⎩−2x +3y =5<br />

⎩2x −3y =7


⎧<br />

⎨ y =−1 2 x +5<br />

⎩x +2y =10<br />

<br />

⎧5x −2y =10<br />

⎨<br />

⎩y = 5 2 x −5<br />

<br />

<br />

⎧2x + y =−4<br />

⎧2x + y =−2<br />

⎨ ⎨<br />

⎩3x −2y =−6<br />

⎩3x − y =7<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −2y =−5<br />

2x −3y =−4<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −3y =−9<br />

2x +5y =4<br />

<br />

⎧<br />

⎨<br />

⎩<br />

5x −2y =−6<br />

y =3x +3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−2x +2y =6<br />

y =−3x +1<br />

<br />

⎧2x +5y =1<br />

⎨<br />

⎩y = 1 3 x −2<br />

<br />

⎧3x +4y =1<br />

⎨<br />

⎩y =− 2 5 x +2<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x + y =5<br />

x −2y = −15<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x + y =10<br />

x −2y = −20<br />

<br />

⎧y =−2x −1<br />

⎨<br />

⎩y =− 1 3 x +4<br />

<br />

⎧<br />

⎨ y = x −6<br />

⎩y =− 3 2 x +4<br />

<br />

⎧ x =2y ⎨<br />

⎩4x −8y =0<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x −16y =8<br />

−x −8y =−4<br />

<br />

⎧<br />

⎨ y = 7 8 x +4<br />

⎩−7x +8y =6<br />

<br />

⎧<br />

⎨ y =−2 3 x +5<br />

⎩2x +3y =11<br />

<br />

<br />

⎧5x +2y =2<br />

⎧6x −5y =−1<br />

⎨ ⎨<br />

⎩−3x − y =0<br />

⎩2x + y =13<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x −5y =7<br />

3x − y =17<br />

<br />

⎧<br />

⎨<br />

⎩<br />

5x −3y =−1<br />

2x − y =2<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x −5y =−9<br />

5x +2y =16<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x −3y =3<br />

2x +5y = −31<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x +8y =−3<br />

2x +5y =−3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

11x +9y =−5<br />

7x +5y =−1<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x +8y =67<br />

5x +3y =60<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x +9y =−4<br />

3x +13y =−7<br />

<br />

⎧<br />

⎨<br />

⎩<br />

1<br />

3 x − y =−3<br />

x + 5 2 y =2<br />

<br />

⎧<br />

⎨ x + 1 2 y = 3 2<br />

1<br />

⎩5 x − 1 5 y =3<br />

<br />

⎧<br />

⎨ x + 1 3 y =−1<br />

1<br />

⎩3 x + 1 <br />

2 y =1<br />

⎧<br />

⎨<br />

⎩<br />

1<br />

3 x − y =−3<br />

<br />

2<br />

3 x + 5 2 y =3<br />

⎧<br />

⎨<br />

⎩<br />

2x + y =3<br />

6x +3y =9<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −4y =−1<br />

−3x +12y =3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−3x − y =8<br />

6x +2y = −16<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x +3y =2<br />

20x +15y =10


a ⎧ −15y = −32<br />

⎨8x a ⎧ =7x −5<br />

⎨y<br />

⎩6x +3y =−5<br />

⎩3x −2y =16<br />

b ⎧ =4y −3<br />

⎨x<br />

⎩4x −2y =−6<br />

b ⎧ −5y = −42<br />

⎨12x<br />

⎩3x +7y = −15<br />

<br />

a ⎧ =4x +9<br />

⎨y<br />

⎩5x −2y = −21<br />

b ⎧ −4y =24<br />

⎨9x<br />

⎩3x +5y = −14<br />

<br />

a ⎧ −15y = −30<br />

⎨14x<br />

⎩7x +2y =10<br />

b ⎧ =9y −11<br />

⎨x<br />

⎩2x −7y = −27<br />

<br />

<br />

<br />

<br />

<br />

⎧5x +4y =10<br />

⎨<br />

⎩2x =3y +27 .<br />

<br />

⎧3x + y =12<br />

⎨ .<br />

⎩x = y −8<br />

⎧ x + y =10<br />

⎨<br />

⎩x − y =6<br />

a b <br />

c


n = <br />

m = <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

m = 9 2 <br />

<br />

<br />

<br />

<br />

<br />

<br />

9 2 −9 2 .<br />

<br />

<br />

<br />

−6.


$10,000 + $40 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

s = <br />

n =


e = <br />

<br />

c =


Let x = the measure of the first angle.<br />

y = the measure of the second angle<br />

<br />

<br />

x + y =90<br />

<br />

x − y =26<br />

<br />

<br />

⎧ x + y =90<br />

⎨<br />

⎩x − y =26<br />

⎧ x + y =90<br />

⎨<br />

x − y =26<br />

___________<br />

⎩<br />

2x = 116<br />

x =58<br />

x =58 <br />

x + y =90<br />

58 + y =90<br />

y =32<br />

<br />

58 + 32 = 90✓<br />

58 − 32 = 26✓


x = <br />

y =


a = <br />

b = <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

3b +10 <br />

<br />

<br />

b =20


L = <br />

W =


D = rt <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

D = r · t <br />

<br />

<br />

65j =78k<br />

1 2


k = j − 1 2<br />

<br />

⎧<br />

⎨ k = j − 1 2<br />

⎩65j =78k<br />

<br />

k = j − 1 2 <br />

j <br />

65j = 78k<br />

65j = 78 ⎛ ⎝ j − 1 ⎞<br />

2⎠<br />

65j = 78j −39<br />

−13j = −39<br />

j = 3<br />

j =3 <br />

k <br />

k = j − 1 2<br />

k =3− 1 2<br />

k = 5 2 or k =21 2<br />

<br />

3 hours ⎛ ⎝65 mph ⎞ ⎠ = 195 miles<br />

<br />

2 1 2 hours ⎛ ⎝ 78 mph⎞ ⎠ = 195 miles<br />

<br />

<br />

<br />

2 1 2 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

( 1 hour) <br />

4


+ c.<br />

<br />

b − c.


s = <br />

c = <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

13.5 + 1.5 = 15 <br />

<br />

15 · 4 = 60 <br />

<br />

13.5 − 1.5 = 12 <br />

<br />

12 · 5 = 60


j = <br />

w =


347 + 18 = 365 <br />

<br />

365 · 3 = 1,095 <br />

<br />

<br />

347 − 18 = 329 <br />

<br />

329 · 3 = 987


−26. <br />

<br />

<br />

−27. −59.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

−45. −89.


4.5 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


4.025(1,562).


a ·12=12a,<br />

c ·7=7c.<br />

<br />

<br />

a = <br />

c = <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

−7.<br />

<br />

a = 521


n = <br />

d = <br />

<br />

<br />

<br />

<br />

<br />

<br />

n(0.05) = 0.05n <br />

d(0.10) = 0.10d.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

d =2n −9


n =36 <br />

<br />

$0.10 = $6.30<br />

$0.05 = $1.80<br />

= $8.10 ✓<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

n = <br />

c =


Number • Value = Total Value<br />

<br />

<br />

<br />

<br />

<br />

−2 <br />

<br />

<br />

<br />

n =16 <br />

<br />

<br />

16 + 4 = 20 ✓<br />

9 · 16 + 2 · 4 = 152 ✓


x = <br />

y = <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

−0.5 <br />

<br />

<br />

x =80 <br />

<br />

<br />

80 + 120 = 200 ✓<br />

0.25(80) + 0.50(120) = 200 ✓<br />

Yes!


I = Prt. <br />

<br />

<br />

<br />

<br />

7.1% <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

s = <br />

b =


s + b = 40,000, <br />

0.08s + 0.03b = 0.071(40,000), <br />

<br />

<br />

6.2% <br />

<br />

<br />

<br />

<br />

<br />

12.5% <br />

<br />

<br />

<br />

10.5% <br />

5.9%. $1,669.68. <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

b = <br />

f =


C(x) = ⎛ ⎝cost per unit ⎞ ⎠ · x + fixed costs<br />

<br />

R(x) = ⎛ ⎝selling price per unit ⎞ ⎠ · x<br />

<br />

C(x) = R(x)


a <br />

b <br />

c <br />

d <br />

<br />

a <br />

<br />

<br />

C(x) = ⎛ ⎝cost per unit ⎞ ⎠ · x + fixed costs<br />

C(x) = 105x + 7000<br />

b <br />

<br />

<br />

<br />

R(x) = ⎛ ⎝selling price per unit ⎞ ⎠ · x<br />

R(x) = 245x<br />

c <br />

<br />

⎧C(x) = 105x + 7000 ⎧y = 105x + 7000<br />

⎨ or ⎨<br />

⎩R(x) = 245x<br />

⎩y = 245x<br />

d


C(x) = R(x)<br />

105x + 7000 = 245x<br />

7000 = 140x<br />

50 = x<br />

<br />

<br />

(50, 12,250).<br />

<br />

<br />

<br />

a <br />

b <br />

c <br />

d <br />

<br />

<br />

<br />

a <br />

b <br />

c <br />

d


$3.80. <br />

<br />

<br />

<br />

<br />

<br />

<br />

$6.20. <br />

<br />

<br />

<br />

<br />

<br />

$1.80 <br />

$1.20 <br />

<br />

$1.40 <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

$0.95. <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

$13.10. <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

$1.79 <br />

$2.49 <br />

<br />

$2.21


10.6% <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

7.25%<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

5.5% <br />

<br />

<br />

<br />

<br />

<br />

<br />

4.5% <br />

<br />

<br />

$2878.50. <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

5.25%


a <br />

<br />

b <br />

c <br />

<br />

<br />

<br />

<br />

a <br />

b <br />

c <br />

<br />

d <br />

<br />

d <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧ s + b = 40,000<br />

⎨<br />

⎩0.08s + 0.03b = 0.071(40,000) .<br />

<br />

<br />

<br />

a <br />

b


5x − 2y + 3z x = −2, y = −4, z = 3.<br />

<br />

<br />

<br />

⎧−2x + y = −11<br />

⎨<br />

.<br />

⎩ x + 3y = 9<br />

<br />

<br />

<br />

⎧7x + 8y = 4<br />

⎨<br />

⎩3x − 5y = 27 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

ax + by = c, <br />

(x, y),


ax + by + cz = d, <br />

(x, y, z) <br />

<br />

<br />

ax + by + cz = d<br />

(x, y, z)


(x, y, z) <br />

<br />

<br />

<br />

(x, y, z).<br />

<br />

<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x − y + z =2<br />

2x − y − z =−6 .<br />

2x +2y + z =−3<br />

a (−2, −1, 3) b (−4, −3, 4)<br />

a


<br />

<br />

a (1, −3, 2) b (4, −1, −5)<br />

⎧<br />

⎨<br />

⎩<br />

3x + y + z = 2<br />

x + 2y + z = −3.<br />

3x + y + 2z = 4<br />

<br />

<br />

a (2, −2, 3) b (−2, 2, 3)<br />

⎧x<br />

− 3y + z = −5<br />

⎨−3x − y − z = 1 .<br />

⎩2x − 2y +3z = 1<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −2y + z =3<br />

2x + y + z =4 .<br />

3x +4y +3z =−1


⎧<br />

⎨<br />

⎩<br />

3x + y − z = 2<br />

2x − 3y − 2z = 1.<br />

⎩4<br />

x − y − 3z = 0<br />

<br />

<br />

⎧4x + y + z = −1<br />

⎨−2x − 2y + z = 2.<br />

⎩2x + 3y − z = 1<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧3x −4z =0<br />

⎨3y +2z =−3.<br />

⎩2x +3y =−5<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x −4z =0 (1)<br />

3y +2z =−3(2)<br />

2x +3y =−5(3)<br />

z <br />

x y x y <br />

x =−4


x =−4 y =1. x =−4 <br />

(−4, 1, −3)<br />

<br />

3x −4z = 0(1)<br />

3(−4) −4(−3) = ? 0<br />

0 = 0 ✓<br />

3y +2z = −3(2)<br />

3(1) +2(−3) = ? −3<br />

−3 = −3 ✓<br />

2x +3y = −5(3)<br />

2(−4) +3(1) = ? −5<br />

−5 = −5 ✓<br />

The solution is (−4, 1, −3).<br />

<br />

<br />

⎧3x − 4z = −1<br />

⎨2y + 3z = 2 .<br />

⎩2x + 3y = 6<br />

<br />

<br />

⎧4x − 3z = −5<br />

⎨3y + 2z = 7 .<br />

⎩3x + 4y = 6<br />

<br />

<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x +2y −3z =−1<br />

x −3y + z =1 .<br />

2x − y −2z =2<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x +2y −3z =−1(1)<br />

x −3y + z =1 (2)<br />

2x − y −2z =2 (3)<br />

z


⎧ x + 2y + 6z = 5<br />

⎨−x + y − 2z = 3.<br />

⎩x<br />

− 4y − 2z = 1<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x − 2y + 3z = 6<br />

4x − 3y + 2z = 0 .<br />

−2x + 3y − 7z = 1<br />

<br />

<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x +2y − z =1<br />

2x +7y +4z =11.<br />

x +3y + z =4<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x +2y − z =1 (1)<br />

2x +7y +4z =11(2)<br />

x +3y + z =4 (3)<br />

<br />

y


y +2z = 3<br />

y = −2z +3<br />

x +2y − z =1<br />

y =−2z +3. x +2(−2z +3)−z = 1<br />

x −4z +6−z = 1<br />

x −5z +6 = 1<br />

x = 5z −5<br />

0=0 <br />

(x, y, z) x =5z −5;y =−2z +3 <br />

<br />

<br />

⎧ x + y − z = 0<br />

⎨2x + 4y − 2z = 6.<br />

⎩3x + 6y − 3z = 9<br />

<br />

<br />

⎧x<br />

− y − z = 1<br />

⎨−x + 2y − 3z = −4.<br />

⎩3x − 2y − 7z = 0<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

y = 2x<br />

2x − y = 0


y.<br />

x =75 <br />

−2x + y =0<br />

x = 75. −2(75) + y = 0<br />

−150 + y = 0<br />

y = 150 student tickets<br />

x + y + z = 250<br />

<br />

x = 75, y = 150. 75 + 150 + z = 250<br />

225 + z = 250<br />

z = 25 child tickets


⎧<br />

⎨<br />

⎩<br />

2x −6y + z =3<br />

3x −4y −3z =2<br />

2x +3y −2z =3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−3x + y + z =−4<br />

−x +2y −2z =1<br />

2x − y − z =−1<br />

a (3,1,3)b (4, 3, 7)<br />

a (−5, −7, 4) b (5, 7, 4)<br />

<br />

⎧y −10z =−8<br />

⎨2x − y =2<br />

⎩x −5z =3<br />

a (7, 12, 2) b (2, 2, 1)<br />

⎧x +3y − z =15<br />

⎪<br />

⎨y = 2<br />

⎪ 3 x −2<br />

⎩x −3y + z =−2<br />

a ⎛ ⎝ −6, 5, 1 ⎞<br />

2⎠ b ⎛ ⎝ 5, 4 3 ,−3⎞ ⎠<br />

<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

5x +2y + z =5<br />

−3x − y +2z =6<br />

2x +3y −3z =5<br />

<br />

⎧<br />

⎨<br />

⎩<br />

6x −5y +2z =3<br />

2x + y −4z =5<br />

3x −3y + z =−1<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x −5y +3z =8<br />

3x − y +4z =7<br />

x +3y +2z =−3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

5x −3y +2z =−5<br />

2x − y − z =4<br />

3x −2y +2z =−7<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x −5y +4z =5<br />

5x +2y + z =0<br />

2x +3y −2z =3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x −3y + z =7<br />

2x −5y −4z =3<br />

3x −2y −2z =−7<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x +8y +2z =−5<br />

2x +5y −3z =0<br />

x +2y −2z =−1<br />

<br />

⎧<br />

⎨<br />

⎩<br />

11x +9y +2z =−9<br />

7x +5y +3z =−7<br />

4x +3y + z =−3<br />

<br />

⎧1<br />

⎪3 x − y − z =1<br />

⎨x + 5 2<br />

⎪<br />

y + z =−2<br />

⎩2x +2y + 1 2 z =−4<br />

<br />

⎧x + 1<br />

⎪ 2 y + 1 2 z =0<br />

⎨1<br />

5 x − 1 5 y + z =0<br />

<br />

⎪<br />

1<br />

⎩3 x − 1 3 y +2z =−1<br />

⎧x + 1 y −2z =−1<br />

⎪ 3<br />

⎨1<br />

3<br />

⎪<br />

x + y + 1 2 z =0<br />

<br />

1<br />

⎩2 x + 1 3 y − 1 2 z =−1<br />

⎧1<br />

⎪3 x − y + 1 2 z =4<br />

⎨2<br />

3<br />

⎪<br />

x + 5 y −4z =0<br />

2<br />

⎩x − 1 2 y + 3 2 z =2<br />

<br />

⎧x +2z =0<br />

⎨4y +3z =−2<br />

⎩2x −5y =3<br />

<br />

⎧2x +5y =4<br />

⎨3y − z =3<br />

⎩4x +3z =−3<br />

<br />

⎧2y +3z =−1<br />

⎨5x +3y =−6<br />

⎩7x + z =1<br />

<br />

⎧3x − z =−3<br />

⎨5y +2z =−6<br />

⎩4x +3y =−8<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x −3y +2z =0<br />

−2x +3y −7z =1<br />

2x −2y +3z =6<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −2y +2z =1<br />

−2x + y − z =2<br />

x − y + z =5


⎧<br />

⎨<br />

⎩<br />

2x +3y + z =12<br />

x + y + z =9<br />

3x +4y +2z =20<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x +4y + z =−8<br />

4x − y +3z =9<br />

2x +7y + z =0<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x +2y + z =4<br />

x + y −2z =3<br />

−2x −3y + z =−7<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x + y −2z =3<br />

−2x −3y + z =−7<br />

x +2y + z =4<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x + y −3z =−1<br />

y − z =0<br />

−x +2y =1<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −2y +3z =1<br />

x + y −3z =7<br />

3x −4y +5z =7<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


3(x + 2) + 4 = 4(2x − 1) + 9.<br />

<br />

0.25p + 0.25(p + 4) = 5.20.<br />

<br />

<br />

x = −2 y = 3: 2x 2 − xy<br />

+ 3y 2 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

m × n. <br />

2×3.


a ⎧ −3y =−1<br />

⎨5x b<br />

⎩y =2x −2<br />

⎧<br />

⎨<br />

⎩<br />

6x −5y +2z =3<br />

2x + y −4z =5<br />

3x −3y + z =−1<br />

<br />

a <br />

y = 2x −2<br />

−2x + y = −2<br />

<br />

<br />

b <br />

<br />

<br />

<br />

a ⎧ ⎨⎧<br />

⎩<br />

⎧<br />

3x + 8y =−3<br />

2x = −5y −3 b ⎨<br />

⎩<br />

2x − 5y + 3z = 8<br />

3x − y + 4z = 7<br />

⎩ x + 3y + 2z = −3<br />

<br />

<br />

a ⎧ ⎨⎧<br />

⎩<br />

11x =−9y − 5<br />

7x + 5y = −1<br />

⎧<br />

b ⎨<br />

⎩<br />

5x − 3y + 2z = −5<br />

2x − y − z = 4<br />

3x − 2y + 2z = −7<br />

<br />

<br />

<br />

<br />

⎡<br />

⎢ 4 −3 3 −1⎤<br />

1 2 −1 2⎥.<br />

⎣−2 −1 3 −4⎦<br />

<br />

|<br />

<br />

4×3


⎡1 −1 2 3<br />

⎤<br />

⎢2 1 −2 1⎥<br />

⎣4 −1 2 0<br />

⎦ .<br />

<br />

<br />

⎡1 1 1 4<br />

⎤<br />

⎢2 3 −1 8⎥<br />

⎣1 1 −1 3<br />

⎦ .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

−3 <br />

−3


a <br />

b <br />

c −2 <br />

<br />

a <br />

⎡<br />

⎢ 6 −5 2<br />

|<br />

2 1 −4<br />

⎣3 −3 1<br />

3⎤<br />

5⎥<br />

−1⎦<br />

b <br />

c −2 <br />

<br />

<br />

a <br />

b <br />

c <br />

⎡ 5 −2 −2<br />

⎢ 4 −1 −4<br />

⎣−2 3 0<br />

−2<br />

⎤<br />

4⎥<br />

−1<br />

⎦<br />

|<br />

<br />

<br />

a <br />

b <br />

c <br />

⎡2 −3 −2<br />

⎢4 1 −3<br />

⎣5 0 4<br />

−4<br />

⎤<br />

2⎥<br />

−1<br />

⎦<br />

|


⎡<br />

⎣ 1 −1 2⎤<br />

4 −8 | 0⎦ .<br />

<br />

−4 <br />

<br />

<br />

⎡1<br />

−1 2<br />

⎤<br />

⎣3 −6 | 2⎦<br />

.<br />

<br />

<br />

⎡ 1 −1 3<br />

⎤<br />

⎣−2 −3 | 2⎦<br />

.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧3x +4y =5<br />

⎨<br />

⎩x +2y =1 .


⎧2x + y = 7<br />

⎨<br />

⎩x<br />

− 2y = 6 .<br />

<br />

<br />

⎧2x + y = −4<br />

⎨<br />

⎩x<br />

− y = −2 .


⎧<br />

⎨<br />

⎩<br />

3x +8y +2z =−5<br />

2x +5y −3z =0 .<br />

x +2y −2z =−1


⎧<br />

⎨<br />

⎩<br />

2x − 5y + 3z = 8<br />

3x − y + 4z = 7 .<br />

⎩ x + 3y + 2z = −3<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−3x + y + z = −4<br />

−x + 2y − 2z = 1.<br />

2x − y − z = −1


⎧x + y +3z =0<br />

⎨x +3y +5z =0.<br />

⎩2x +4z =1<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

0≠1 <br />

<br />

<br />

<br />

⎧x<br />

− 2y + 2z = 1<br />

⎨−2x + y − z = 2.<br />

⎩x<br />

− y + z = 5<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x + 4y − 3z = −2<br />

2x + 3y − z =−12.<br />

⎩ x + y − 2z = 6


⎧<br />

⎨<br />

⎩<br />

x −2y +3z =1<br />

x + y −3z =7 .<br />

3x −4y +5z =7<br />

<br />

<br />

<br />

<br />

<br />

−2 <br />

<br />

<br />

0=0 <br />

<br />

<br />

<br />

y =2z +2.<br />

<br />

<br />

<br />

(x, y, z), x = z +5;y =2z +2;z


⎧ x + y − z = 0<br />

⎨2x + 4y − 2z = 6.<br />

⎩3x + 6y − 3z = 9<br />

<br />

<br />

⎧x<br />

− y − z = 1<br />

⎨−x + 2y − 3z = −4.<br />

⎩3x − 2y − 7z = 0


a ⎧ − y =−1<br />

⎨3x a ⎧ +4y =−5<br />

⎨2x a ⎧ − y =−4<br />

⎨3x<br />

⎩2y =2x +5<br />

⎩3x −2y =2<br />

⎩2x = y +2<br />

⎧4x +3y =−2<br />

⎧3x −2y − z =−2<br />

⎧x −3y −4z =−2<br />

b ⎨x −2y −3z =7<br />

b ⎨−2x + y =5<br />

b ⎨4x +2y +2z =5<br />

⎩2x − y +2z =−6<br />

⎩5x +4y + z =−1<br />

⎩2x −5y +7z =−8<br />

<br />

a ⎧ −5y =−3<br />

⎨2x<br />

⎩4x =3y −1<br />

⎧<br />

b ⎨<br />

⎩<br />

4x +3y −2z =−3<br />

−2x + y −3z =4<br />

−x −4y +5z =−2<br />

<br />

⎡ ⎣ 2 −1 4⎤<br />

1 −3 | 2⎦ ⎡ ⎣ 2 −4 −2⎤<br />

3 −3 | −1⎦<br />

|<br />

<br />

⎡1 0 −3<br />

|<br />

⎢1 −2 0<br />

⎣0 −1 2<br />

−1⎤<br />

−2⎥<br />

3⎦<br />

<br />

⎡2 −2 0<br />

⎢0 2 −1<br />

⎣3 0 −1<br />

−1⎤<br />

2⎥<br />

−2⎦<br />

<br />

<br />

⎡ ⎣ 6 −4 3⎤<br />

3 −2 | 1⎦<br />

⎡ ⎣ 4 −6 −3⎤<br />

⎡4 −12 −8<br />

3 2 |<br />

|<br />

16⎤<br />

1⎦<br />

⎢4 −2 −3 −1⎥<br />

⎣−6 2 −1 −1⎦<br />

a a <br />

a <br />

b b <br />

b <br />

c −2 c <br />

<br />

<br />

c −2 <br />

<br />

⎡<br />

⎢ 6 −5 2<br />

|<br />

2 1 −4<br />

⎣3 −3 1<br />

3⎤<br />

5⎥<br />

−1⎦<br />

a <br />

b <br />

c −2 <br />

<br />

<br />

<br />

<br />

<br />

⎡<br />

⎣ 1 2 5⎤<br />

−3 −4 | −1⎦ ⎡1 −2 3<br />

|<br />

−4⎤<br />

⎢3 −1 −2 5⎥.<br />

⎣2 −3 −4 −1⎦


⎧2x + y =2<br />

⎧3x + y =2<br />

⎨ ⎨<br />

⎩x − y =−2<br />

⎩x − y =2<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−x +2y =−2<br />

x + y =−4<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−2x +3y =3<br />

x +3y =12<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x −3y + z =19<br />

−3x + y −2z = −15<br />

x + y + z =0<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x − y +3z =−3<br />

−x +2y − z =10<br />

x + y + z =5<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x −6y + z =3<br />

3x +2y −3z =2<br />

2x +3y −2z =3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x −3y + z =7<br />

2x −5y −4z =3<br />

3x −2y −2z =−7<br />

<br />

⎧x +2z =0<br />

⎨4y +3z =−2<br />

⎩2x −5y =3<br />

<br />

⎧2x +5y =4<br />

⎨3y − z =3<br />

⎩4x +3z =−3<br />

<br />

⎧2y +3z =−1<br />

⎨5x +3y =−6<br />

⎩7x + z =1<br />

<br />

⎧3x − z =−3<br />

⎨5y +2z =−6<br />

⎩4x +3y =−8<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x +3y + z =12<br />

x + y + z =9<br />

3x +4y +2z =20<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x +2y +6z =5<br />

−x + y −2z =3<br />

x −4y −2z =1<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x +2y −3z =−1<br />

x −3y + z =1<br />

2x − y −2z =2<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x −3y +2z =0<br />

−2x +3y −7z =1<br />

2x −2y +3z =6<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x − y +2z =−4<br />

2x + y +3z =2<br />

−3x +3y −6z =12<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−x −3y +2z =14<br />

−x +2y −3z =−4<br />

3x + y −2z =6<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x + y −3z =−1<br />

y − z =0<br />

−x +2y =1<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x +2y + z =4<br />

x + y −2z =3<br />

−2x −3y + z =−7<br />

<br />

⎧ x + y =10<br />

⎨<br />

⎩x − y =6<br />

a <br />

b c <br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x + y =12<br />

x = y −8


a <br />

b


2×2 <br />

3×3 <br />

<br />

<br />

<br />

<br />

5(−2) − (−4)(1).<br />

<br />

−3(8 − 10) + (−2)(6 − 3) − 4 ⎛ ⎝−3 − (−4) ⎞ ⎠.<br />

<br />

<br />

−12<br />

−8 .<br />

<br />

<br />

<br />

2×2 <br />

<br />

⎡ ⎣ a b ⎤<br />

c d⎦ , <br />

|<br />

a b<br />

c<br />

d| . <br />

<br />

⎡ ⎣ a<br />

c<br />

<br />

a ⎡ ⎣ 4 −2 ⎤<br />

3 −1⎦ b ⎡ ⎣ −3 −4 ⎤<br />

−2 0⎦ .<br />

<br />

a<br />

b ⎤<br />

d⎦ , <br />

|<br />

a b<br />

= ad − bc<br />

c d|


<br />

<br />

<br />

<br />

<br />

a ⎡5<br />

⎣ −3<br />

2 −4⎦<br />

⎤ b ⎡−4<br />

⎣ −6<br />

⎤<br />

0 7⎦<br />

.<br />

<br />

a ⎡−1 ⎣ 3<br />

−2 4⎦<br />

⎤ b ⎡−7<br />

⎣ −3<br />

⎤<br />

−5 0⎦<br />

.<br />

3×3 <br />

3×3 <br />

2×2 3×3 <br />

<br />

3×3 <br />

3×3 2×2 <br />

3×3 <br />

a 1 , <br />

2×2 <br />

b 2 , <br />

2×2


a<br />

<br />

|<br />

4<br />

−2 3<br />

1 0 −3<br />

−2 −4 2|, a a 1 b b 3 c c 2 .<br />

a 1 .<br />

2×2 <br />

<br />

<br />

b<br />

b 3 .<br />

2×2 <br />

<br />

<br />

c<br />

c 2 .


2×2 <br />

<br />

<br />

<br />

<br />

|<br />

1<br />

−1 4<br />

0 2 −1<br />

−2 −3 3|, a a 1 b b 2 c c 3 .<br />

<br />

<br />

|−2 −1 0<br />

3 0 −1 a a 2 b b 3 c c 2 .<br />

−1 −2 3|,<br />

3×3 <br />

3×3 2×2 <br />

3×3 <br />

<br />

3×3 <br />

3×3 <br />

<br />

<br />

|<br />

2 −3 −1<br />

3 2 0 <br />

−1 −1 −2|


|3 −2 4<br />

0 −1 −2 <br />

2 3 −1|,<br />

<br />

<br />

|<br />

3 −2 −2<br />

2 −1 4 <br />

−1 0 −3|,<br />

3×3 <br />

<br />

<br />

<br />

|+ − +<br />

+|<br />

− + −<br />

+ −<br />

<br />

<br />

|+ − +<br />

+|<br />

− + −<br />

+ −


|4 −1 3 0 2 <br />

5 −4 −3|<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

|2 −1 0 3 −4 <br />

3 −4 −3|<br />

<br />

<br />

|−2 −1 −3<br />

0|<br />

−1 2 2 <br />

4 −4<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧ a 1 x + b 1 y = k 1<br />

⎨ , (x, y) <br />

⎩a 2 x + b 2 y = k 2


D x D y ,<br />

<br />

<br />

<br />

⎧2x + y =−4<br />

⎨<br />

⎩3x −2y =−6 .


⎧<br />

⎨<br />

⎩<br />

3x + y = −3<br />

2x + 3y = 6 .<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−x + y = 2<br />

2x + y = −4 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

D x . <br />

D y . <br />

x = D x<br />

D ,<br />

y = D y<br />

D<br />

<br />

<br />

<br />

<br />

3×3 <br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

a 1 x + b 1 y + c 1 z = k 1<br />

a 2 x + b 2 y + c 2 z = k 2 , (x, y, z) <br />

a 3 x + b 3 y + c 3 z = k 3


⎧<br />

⎨<br />

⎩<br />

3x −5y +4z =5<br />

5x +2y + z =0 .<br />

2x +3y −2z =3<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

D x . <br />

<br />

<br />

<br />

<br />

<br />

D y .


D z . <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

(2, −3, −4).<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x + 8y + 2z = −5<br />

2x + 5y − 3z = 0 .<br />

⎩ x + 2y − 2z = −1<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x + y − 6z = −3<br />

2x + 6y + 3z = 0 .<br />

3x + 2y − 3z = −6<br />

<br />

D =0, <br />

D =0 D x , D y D z <br />

<br />

D =0 D x , D y D z <br />

<br />

D =0,<br />

Value of determinants Type of system Solution<br />

D = 0 and D x , D y and D z are all zero consistent and dependent infinitely many solutions<br />

D = 0 and D x , D y and D z are not all zero inconsistent no solution


⎧<br />

⎨<br />

⎩<br />

x +3y =4<br />

−2x −6y =3 .<br />

<br />

Evaluate the determinant D, using the<br />

coefficients of the variables.<br />

⎧ x +3y =4<br />

⎨<br />

⎩−2x −6y =3<br />

1 3<br />

D =<br />

| −2 −6|<br />

D =−6−(−6)<br />

D =0<br />

D x D y ,<br />

<br />

Evaluate the determinant D x . D x =<br />

| 4 3<br />

3 −6|<br />

D x =−24−9<br />

D x =15<br />

<br />

<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x − 3y = 8<br />

8x − 6y = 14 .<br />

<br />

<br />

⎧ x = −3y + 4<br />

⎨<br />

⎩2x + 6y = 8 .<br />

<br />

(x 1 , y 1 ),<br />

(x 3 , y 3 ) <br />

(x 2 , y 2 ) <br />

<br />

(x 1 , y 1 ),<br />

|x 1 y 1<br />

1|<br />

1<br />

x 2 y 2 1<br />

x 3 y 3<br />

=0<br />

(x 2 , y 2 ) (x 3 , y 3 ) <br />

<br />

|x 1 y 1<br />

1|<br />

1<br />

x 2 y 2 1<br />

x 3 y 3<br />

=0<br />

<br />

(5, −5),<br />

(4, −3), (3, −1)


(5, −5), (4, −3), (3, −1)<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

(3, −2), (5, −3), (1, −1) <br />

(−4, −1), (−6, 2), (−2, −4)


⎡ ⎣ 6 −2 ⎤<br />

3 −1⎦ ⎡ ⎣ −4 8 ⎤<br />

−3 5⎦<br />

⎡ ⎣ −3 5 ⎤<br />

0 −4⎦<br />

⎡ ⎣ −2 0 ⎤<br />

7 −5⎦<br />

<br />

<br />

|<br />

3 −1<br />

5|<br />

4<br />

|−1 −3 2<br />

−1 0 −2<br />

4 −2 −1<br />

−4 1 −2 0 −3|<br />

a a 1 b b 2 c c 3<br />

<br />

|<br />

2 −3 −4<br />

−1 2 −3<br />

0 −1 −2|<br />

<br />

a a 2 b b 2 c c 2<br />

a a 1 b b 1 c c 2<br />

|−2 −2 3<br />

1 −3 0<br />

−2 3 −2|<br />

a a 3 b b 3 c c 3<br />

<br />

|−2 3 −1<br />

−1 2 −2<br />

<br />

3 1 −3| |<br />

4 −1 −2<br />

7|<br />

−3 −2 1<br />

−2 −5<br />

<br />

|−2 −3 −4<br />

0|<br />

5 −6 7<br />

<br />

−1 2<br />

|1 3 −2<br />

5 −6 4<br />

0 −2 −1|<br />

<br />

|−5 −1 −4<br />

6| |<br />

4 −1 3<br />

4|<br />

4 0 −3<br />

3 −2 2<br />

2 −2 −1 0<br />

|<br />

3 5 4<br />

1|<br />

|2 −4 −3<br />

0|<br />

−1 3 0<br />

5 −1 −4<br />

−2 6 3 2<br />

<br />

<br />

⎧−2x +3y =3<br />

⎧x −2y =−5<br />

⎨ ⎨<br />

⎩x +3y =12<br />

⎩2x −3y =−4<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −3y =−9<br />

2x +5y =4<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x + y =−4<br />

3x −2y =−6<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −2y =−5<br />

2x −3y =−4<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −3y =−9<br />

2x +5y =4


⎧5x −3y =−1<br />

⎨<br />

⎩2x − y =2<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x +8y =−3<br />

2x +5y =−3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

6x −5y +2z =3<br />

2x + y −4z =5<br />

3x −3y + z =−1<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x −3y + z =7<br />

2x −5y −4z =3<br />

3x −2y −2z =−7<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x −5y +3z =8<br />

3x − y +4z =7<br />

x +3y +2z =−3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

11x +9y +2z =−9<br />

7x +5y +3z =−7<br />

4x +3y + z =−3<br />

<br />

⎧x +2z =0<br />

⎨4y +3z =−2<br />

⎩2x −5y =3<br />

<br />

⎧2x +5y =4<br />

⎨3y − z =3<br />

⎩4x +3z =−3<br />

<br />

⎧2y +3z =−1<br />

⎨5x +3y =−6<br />

⎩7x + z =1<br />

<br />

⎧3x − z =−3<br />

⎨5y +2z =−6<br />

⎩4x +3y =−8<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x + y =3<br />

6x +3y =9<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −4y =−1<br />

−3x +12y =3<br />

<br />

⎧<br />

⎨<br />

⎩<br />

−3x − y =4<br />

6x +2y = −16<br />

<br />

⎧<br />

⎨<br />

⎩<br />

4x +3y =2<br />

20x +15y =5<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x + y −3z =−1<br />

y − z =0<br />

−x +2y =1<br />

<br />

⎧<br />

⎨<br />

⎩<br />

2x +3y + z =12<br />

x + y + z =9<br />

3x +4y +2z =20<br />

<br />

⎧<br />

⎨<br />

⎩<br />

3x +4y −3z =−2<br />

2x +3y − z = −12<br />

x + y −2z =6<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −2y +3z =1<br />

x + y −3z =7<br />

3x −4y +5z =7<br />

<br />

<br />

(0, 1), (2, 0), (−2, 2). (0, −5), (−2, −2), (2, −8).<br />

(4, −3), (6, −4), (2, −2). (−2, 1), (−4, 4), (0, −2).<br />

<br />

<br />

<br />

<br />

3×3 <br />

<br />

<br />

<br />

<br />

<br />

a <br />

b


2a < 5a + 12.<br />

<br />

<br />

⎛ ⎝ ⎛ 3, 1<br />

2<br />

⎞<br />

⎠<br />

⎞ y > 2x + 3.<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧x +4y ≥10<br />

⎨<br />

⎩3x −2y


(−2, 4) (−2, 4) <br />

b (3, 1) <br />

(3, 1) (3, 1) <br />

<br />

<br />

<br />

a (3, −1) b (6, −3)<br />

⎧x<br />

− 5y > 10<br />

⎨<br />

⎩2x + 3y > −2 .<br />

<br />

<br />

a (−2, 1) b (4, −1)<br />

⎧ y > 4x − 2<br />

⎨<br />

⎩4x − y < 20 .<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧y ≥2x −1<br />

⎨<br />

⎩y < x +1 .


⎧ y < 3x + 2<br />

⎨<br />

⎩ y > −x − 1 .<br />

<br />

<br />

⎧ y < − 1x + 3<br />

⎨ 2 .<br />

⎩ y < 3x − 4<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

⎧x − y >3<br />

⎨<br />

⎩y 3<br />

⎨<br />

⎩y


y −1 4 x + 5.<br />

<br />

<br />

⎧<br />

⎨<br />

⎩<br />

x −2y −4


⎧<br />

⎨<br />

⎩<br />

x −2y −4<br />

x −2y −4, y =−4 <br />

<br />

y =−4. <br />

<br />

<br />

<br />

(0, 0) <br />

<br />

<br />

<br />

<br />

⎧y<br />