Math 017 Materials With Exercises

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$Math 171 Review for the Final Exam$

$Math 171 Test 2$

$MATH 171 SYLLABUS$

) This should be written as 4x 2 (3x 2)Notice that “subtract from”causes us to reverse the order of the expressions. Removing parentheses following aminus sign reverses signs of all terms inside parentheses ( 3 x „becomes‟ 3x, 2„becomes‟ 2 ). After removing parentheses we collect like terms ( x's separately,numbers separately) to get 4x 2 (3x 2) 4x 2 3x 2 7x 4c) This should be written as ( 3x 2)( 4x 2). To remove parentheses we applythe Distributive Law ( 3x 2)( 4x 2) 3x(4x) 3x(2) 2( 4x) 2(2)12x2 6x 8x 4 12x214x 455Example 6.5 Rewrite the expression 3( x 2) (4 x ) in its equivalent form without parenthesesand simplify by collecting like terms.Solution:3( x5 2) (4 x5) 3x5 (3)( 2) 4 x5 3x5 x5 6 4 4x510The square of the sum or the difference of two expressionsThe Distributive Law, together with the ability to collect like terms, allows us to derive two importantformulas.2Consider ( a b) . Our goal is to write the expression in its equivalent form without parentheses.Thus the following is true.( a b)2 use the definition of the square of the expression( a b)(a b)apply the Distributive Law for the product of two sums22a ab ba b collect like terms22a 2ab bThe Square of the Sumof Two Expressions.2 22( a b) a 2ab bNotice that this means that the square of the sum of two terms is not equal to the sum of their2 2 2squares. In other words ( a b) a b .Using exactly the same technique, one can prove thatThe Square of the Differenceof Two Expressions.2 22( a b) a 2ab bAgain, this means that the square of the difference of two terms is not equal to the difference of2 2 2their squares. In other words ( a b) a b .64
Example 6.6 Rewriteterms.( 2 x2 ) in its equivalent form without parentheses. Simplify by collecting likeSolution:22(2 x) (2 x)(2 x) 22 2( x) x(2) x(x) 4 2x 2x x 4 4x x2Common mistakes and misconceptionsMistake 6.122 apples 3apples 5(apples) . Similarly, 2xxInstead,2x 3x 5x(although 2 x2x(3x) 6 ).2 3x 5 .Mistake 6.23 x ( a b) 3x a bThe minus sign pertains to the whole expression ( a b), and all signs must be „changed to theopposite‟, not only the first one, 3 x ( a b) 3x a b .Mistake 6.3Although it is true that( b2 2 2ab) a ,2 2 2( a b) a bMistake 6.4x 2xWhen collecting like terms in you CANNOT multiply the expression by 3 to get rid of3 3x 2x1 2fractions. x 2x(it would be like saying that 12).3 33 3<strong>Exercises</strong> with Answers (For answers see Appendix A)Ex.1Which of the following rows consists of all like terms?a) 1 22 1 x,4x, 0.5x, x2 , 3x263b) 3xy,8yx, 0.6xy,2xy, xy72 3 4c) 5x, 5x, 5x, 5x, 5d) 2a, 5a,7a,29a,a,11aEx.2 Are x and x like terms?Ex.3 Are any of the following like terms: 7 x 2 y ,27xy , and27(xy ) ?65
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Math 017MaterialsWithExercises
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TABLE OF CONTENTSLesson 1Variables
Page 5 and 6:
Lesson 1___________________________
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e) „minus x ‟ or „the opposit
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For example,Evaluate 5 x when x 10
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3xd) x( y)e) 2f) 2x yz w(t)Ex.4 T
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Ex.16 Let x 3. Rewrite the express
Page 15 and 16: e) x5f) x2Ex.28 Substitute“undefi
Page 17 and 18: Lesson 2___________________________
Page 19: Solution:Please, notice the use of
Page 22 and 23: g)i)4( a 2)h)8y (x)4(ab)j) ( a d)
Page 24 and 25: 1 2Ex.19 If possible, evaluate when
Page 26 and 27: Lesson 3___________________________
Page 28 and 29: Commutative Propertyof Additionx y
Page 30 and 31: 2 x 2 xFor example, and thus als
Page 32 and 33: Example 3.10 Determine which of the
Page 34 and 35: Exercises with Answers (For answers
Page 36 and 37: Ex.16 Write the following expressio
Page 38 and 39: s) ( 0.2xy)( 0.3z)t)4x1Ex.24 Is 1
Page 40 and 41: Lesson 4___________________________
Page 42 and 43: Numerical coefficientsNumerical Coe
Page 44 and 45: For your convenience, we will displ
Page 46 and 47: 9 818081a) 9 98097 6 76134 4 4 4
Page 48 and 49: Ex.8Simplify by raising to the indi
Page 50 and 51: 73a 2xxi) 2aj)2( 42a)x(3xk)4x2 ) 3
Page 52 and 53: Lesson 5___________________________
Page 54 and 55: FactorizationThe Distributive Law l
Page 56 and 57: Simplification of algebraic fractio
Page 58 and 59: Mistake 5.1When factoring 3 from 3x
Page 60 and 61: Ex.13 Factor2a) 5 a from the follow
Page 62 and 63: 24x ( 5x)g)xa bi)7(a b)bc ( b e)
Page 64 and 65: Adding and subtracting like termsIf
Page 68 and 69: Ex.4 Circle all terms that are like
Page 70 and 71: 682 122k) 6 ( d a) a dl) 3a1(4a
Page 72 and 73: Lesson 7___________________________
Page 74 and 75: For instance, the area of a rectang
Page 76 and 77: a) a b 2bc) a bd) ae)b)aba 22a a
Page 78 and 79: Ex.24 Express the expressiona) m s
Page 80 and 81: A solution, the solution set, and w
Page 82 and 83: ■ Solve the following equation 3x
Page 84 and 85: sides (subtracting 4 x from both si
Page 86 and 87: Mistake 8.2While solving an equatio
Page 88 and 89: Ex.12 Determine whether the followi
Page 90: Lesson 9___________________________
Page 94 and 95: xy x 1 yFactor x .x( y 1)1yDivide
Page 96 and 97: a) b b) a c) tEx.15. SolveEx.16. So
Page 98 and 99: For each number you should check if
Page 100 and 101: ■ Solve the inequality 4x 9 1.
Page 102 and 103: ■ Solve the following inequality
Page 104 and 105: Ex.10 Graph the following number se
Page 106 and 107: d) z 3e) z 13 Ex.22 Knowing x 1
Page 108 and 109: Example 11.2 The expression( 22 2x
Page 110 and 111: Solution:a) To match the form y mx
Page 112 and 113: a x and b 3 , so now we can rewri
Page 114 and 115: Ex.9 For each of the following expr
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114xc) y 5d) y 3x 2 0e) 3y 6x1f
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APPENDIX A:Lesson 1ANSWERS TO EXERC
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49C 3255 2(L+W)62mc75a) a b expon
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3(m n)a(x1)g)h)2tx19s s s sThe opp
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52233x1 2 x5 3 1513 a) m ( 2m) 2mb
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c) Numerator: one term, 3ab (with t
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18 a) 9 b) 3 c) 3 d) 3e) 6 f) 279
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11 a) x 1 b)12 a) x abcx a b42 b
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c) x 2d) a 7e) x 10f) a 225 a)
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x or 4x ( 7) 0; a 4; b 7g) x (
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3. Let m = ─3. Evaluate the follo
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1. Write the following statements a
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6. Solve the following equations:8.
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8. Factor the following expressions
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3 ;b) x 3 y 6 xy 2 A xy 28. a) x
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Math 017 Materials With Exercises

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