Beginning and Intermediate Algebra - Wallace Math Courses ...

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The perimeter of a rectangle is 44. The length is 5 less than double the width. Find the dimensions. Lengthx We will make the length x Width 2x − 5 Width is five less than two times the length P =2L + 2W The formula for perimeter of a rectangle (44) =2(x)+2(2x − 5) Replace P , L, and W with labeled values 44 = 2x +4x − 10 Distribute through parenthesis 44 =6x − 10 Combine like terms 2x +4x + 10 + 10 Add 10 to both sides 54 =6x The variable is multiplied by 6 6 6 Divide both sides by 6 9=x Our solution for x Length 9 Replace x with9in the origional list of sides Width 2(9) − 5 = 13 The dimensions of the rectangle are 9 by 13. We have seen that it is imortant to start by clearly labeling the variables in a short list before we begin to solve the problem. This is important in all word problems involving variables, not just consective numbers or geometry problems. This is shown in the following example. Example 109. A sofa and a love seat together costs S444. The sofa costs double the love seat. How much do they each cost? Love Seatx With no information about the love seat, this is ourx Sofa 2x Sofa is double the love seat, so we multiply by 2 S +L=444 Together they cost 444, so we add. (x) +(2x)=444 Replace S and L with labeled values 3x = 444 Parenthesis are not needed, combine like termsx+2x 3 3 Divide both sides by 3 x = 148 Our solution forx Love Seat 148 Replace x with 148 in the origional list Sofa 2(148)=296 The love seat costs S148 and the sofa costs S296. Be careful on problems such as these. Many students see the phrase “double” and believe that means we only have to divide the 444 by 2 and get S222 for one or both of the prices. As you can see this will not work. By clearly labeling the variables in the original list we know exactly how to set up and solve these problems. 68
1.8 Practice - Number and Geometry Problems Solve. 1. When five is added to three more than a certain number, the result is 19. What is the number? 2. If five is subtracted from three times a certain number, the result is 10. What is the number? 3. When 18 is subtracted from six times a certain number, the result is − 42. What is the number? 4. A certain number added twice to itself equals 96. What is the number? 5. A number plus itself, plus twice itself, plus 4 times itself, is equal to − 104. What is the number? 6. Sixty more than nine times a number is the same as two less than ten times the number. What is the number? 7. Eleven less than seven times a number is five more than six times the number. Find the number. 8. Fourteen less than eight times a number is three more than four times the number. What is the number? 9. The sum of three consecutive integers is 108. What are the integers? 10. The sum of three consecutive integers is − 126. What are the integers? 11. Find three consecutive integers such that the sum of the first, twice the second, and three times the third is − 76. 12. The sum of two consecutive even integers is 106. What are the integers? 13. The sum of three consecutive odd integers is 189. What are the integers? 69
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Beginning and Intermediate Algebra
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Special thanks to: My beautiful wif
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Chapter 6: Factoring 6.1 Greatest C
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0.1 Pre-Algebra - Integers Objectiv
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Multiplication and division of inte
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45) (4)( − 6) Find each quotient.
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9 ÷ 3 3 = 21 ÷ 3 7 Our Soultion T
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Example 21. 13 6 − 9 6 4 6 2 3 Sa
Page 17 and 18: Find each quotient. 37) − 2 ÷ 7
Page 19 and 20: the expression that is to be evalua
Page 21 and 22: Solve. 1) − 6 · 4( − 1) 3) 3+(
Page 23 and 24: It will be more common in our study
Page 25 and 26: 0.4 Practice - Properties of Algebr
Page 27 and 28: Chapter 1 : Solving Linear Equation
Page 29 and 30: Addition Problems To solve equation
Page 31 and 32: 8x = − 24 8 8 x = − 3 Division
Page 33 and 34: 1.2 Linear Equations - Two-Step Equ
Page 35 and 36: − 1 − 1 Divide both sides by
Page 37 and 38: 1.3 Solving Linear Equations - Gene
Page 39 and 40: Example 63. − 3x +9=6x − 27 Not
Page 41 and 42: 4[ − 2]+9=−15 +8(2) Finish mult
Page 43 and 44: 1.4 Solving Linear Equations - Frac
Page 45 and 46: Example 72. � � 3 5 4 x + = 3 2
Page 47 and 48: 1.5 Solving Linear Equations - Form
Page 49 and 50: Example 79. A = πr 2 +πrs for s S
Page 51 and 52: 1.5 Practice - Formulas Solve each
Page 53 and 54: |x| = 5 Absolute value can be posit
Page 55 and 56: Example 90. 7+|2x − 5| =4 Notice
Page 57 and 58: 1.7 Solving Linear Equations - Vari
Page 59 and 60: Once we have found the constant of
Page 61 and 62: 1.7 Practice - Variation Write the
Page 63 and 64: sure. If the pressure of a certain
Page 65 and 66: Fifteen more than three times a num
Page 67: When we started with our first, sec
Page 71 and 72: 30. The perimeter of a college bask
Page 73 and 74: Our equation comes from the future
Page 75 and 76: 4x + 2=102 Solve the two − step e
Page 77 and 78: years ago, the bronze plaque was on
Page 79 and 80: 1.10 Solving Linear Equations - Dis
Page 81 and 82: Rate Time Distance Bob r + 2 3 Fred
Page 83 and 84: 2 =t Our solution fort, she catches
Page 85 and 86: Find the distance he rode. 8. A man
Page 87 and 88: The first plan is flying 25 mph slo
Page 89 and 90: 2.1 Graphing - Points and Lines Obj
Page 91 and 92: E B D F G C A Our Solution The main
Page 93 and 94: x y − 3 − 4 0 − 2 3 0 Our com
Page 95 and 96: 2.2 Graphing - Slope Objective: Fin
Page 97 and 98: When mathematicians began working w
Page 99 and 100: Example 131. Find the value of x be
Page 101 and 102: 9) 10) Find the slope of the line t
Page 103 and 104: y = − 2 x + 3 Our Solution 3 Iden
Page 105 and 106: 2.3 Practice - Slope-Intercept Writ
Page 107 and 108: 2.4 Graphing - Point-Slope Form Obj
Page 109 and 110: Find the equation of the line throu
Page 111 and 112: Write the point-slope form of the e
Page 113 and 114: m = 2 5 m = 2 5 Example 147. Slope
Page 115 and 116: 2.5 Practice - Parallel and Perpend
Page 117 and 118: Chapter 3 : Inequalities 3.1 Solve
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than 4. If we have an expression su
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− 5 − 5 − 2x � 6 Divide bot
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Solve each inequality, graph each s
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As the graphs overlap, we take the
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3.2 Practice - Compound Inequalitie
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edition of a college algebra text!
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It is important to remember as we a
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Chapter 4 : Systems of Equations 4.
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ested in the point that is a soluti
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that parallel lines have the same s
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4.2 Systems of Equations - Substitu
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tion. This process is described and
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World View Note: French mathematici
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31) 2x + y = 2 3x + 7y = 14 33) x +
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6x +9y = 6 New second equation −
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Just as with graphing and substutio
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4.4 Systems of Equations - Three Va
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just the ordered pair (x, y). In th
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15x − 12y +9z = − 12 5x − 4y
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23) 3x + 3y − 2z = 13 6x + 2y −
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are given to use. This means someti
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− 0.5 − 0.5 c=17 We have c, num
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− 0.06x − 0.06y = − 240 Add e
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14) A bank contains 27 coins in dim
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4.6 Systems of Equations - Mixture
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Amount Part Total Start 40 3 120 Ad
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4(30 −n)+2.5n=105 Substitute into
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16) A certain grade of milk contain
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175
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5.1 Polynomials - Exponent Properti
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A quicker method to arrive at the s
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Example 210. Example 211. 7a 3 (2a
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5.2 Polynomials - Negative Exponent
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As we simplified our fraction we to
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5.2 Practice - Negative Exponents S
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Example 223. Convert 3.21 × 10 5 t
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5.3 Practice - Scientific Notation
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World View Note: Ada Lovelace in 18
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21) (3+b 4 )+(7+2b + b 4 ) 22) (1+6
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Just as we distribute a monomial th
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Example 243. (2x − 5)(4x 2 − 7x
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5.6 Polynomials - Multiply Special
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Be very careful when we are squarin
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5.7 Polynomials - Divide Polynomial
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3. Change the sign of the terms and
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Example 261. 2x 3 + 42 − 4x x + 3
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Chapter 6 : Factoring 6.1 Greatest
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ers using mental math, then we take
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6.1 Practice - Greatest Common Fact
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Example 273. 10ab + 15b + 4a+6 Spli
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example the binomials are (a + b) a
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6.3 Factoring - Trinomials where a
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As the past few examples illustrate
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6.3 Practice - Trinomials where a =
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Example 296. 10x 2 − 27x +5 Multi
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6.5 Factoring - Factoring Special P
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Example 306. 4x 2 + 20xy + 25y 2 Mu
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6.5 Practice - Factoring Special Pr
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Example 313. Example 314. Example 3
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6.7 Factoring - Solve by Factoring
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0=(2x +3)(2x − 3) One factor must
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6.7 Practice - Solve by Factoring S
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7.1 Rational Expressions - Reduce R
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However, if there is more than just
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41) 8m+16 20m − 12 43) 2x2 − 10
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Example 333. 25x2 24y4 · 9y8 55x7
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Simplify each expression. 1) 8x2 9
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7.3 Rational Expressions - Least Co
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As the above example illustrates, w
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7.4 Rational Expressions - Add & Su
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13 12 Our Solution The same process
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7.4 Practice - Add and Subtract Add
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2(12) 1(12) − 3 4 5(12) 1(12) + 6
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LCD =(x + 3)(x − 3) Multiply each
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23) 25) 27) 29) x − 4+ 9 2x +3 x
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(20)(9)=6x Multiply 180 = 6x Divide
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3.5 6 The man hasashadow of 3.5 fee
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33) Kali reduced the size of a pain
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LCD will be more involved. We will
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x=5 or − 2 Our Solution World Vie
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7.8 Rational Expressions - Dimensio
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� 435 1 �� 1 lbs = 16 435
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To focus on the process of conversi
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7.8 Practice - Dimensional Analysis
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Chapter 8 : Radicals 8.1 Square Roo
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√ process is being able to transl
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Simplify. √ 1) 245 √ 3) 36 √
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Example 384. 3√ 54 We are working
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8.3 Radicals - Adding Radicals Obje
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Simiplify 1) 2 5 √ + 2 5 √ + 2
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√ √ √ √ 4 50 + 6 30 − 8 3
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The previous example could have bee
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8.5 Radicals - Rationalize Denomina
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2x2 √ √ 10 − 6x 3x Our Soluti
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Example 408. 2 5 √ − 3 7 √ 5
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37) 5 2 √ + 3 √ 5+5 2 √ 38) 3
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1 (3) 4 Evaluate exponent 1 81 Our
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8.6 Practice - Rational Exponents W
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Example 420. ab2 3√ a2 4√ b Rew
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8.7 Practice - Radicals of Mixed In
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Example 426. Example 427. i 35 Divi
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√ Dividing with complex numbers a
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Simplify. 1) 3 − ( − 8+4i) 3) (
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Chapter 9 : Quadratics 9.1 Solving
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Example 444. + 1 +1 Add1to both sid
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2 − 2=0 Subtract 0=0 True! It wor
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Solve. √ 1) 2x + 3 √ 3) 6x −
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x4 4√ = ± 4√ 16 Simplify roots
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5√ 4x +1 = ± 3 Clear radical by
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9.3 Quadratics - Complete the Squar
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x2 + 5 25 x+ 3 36 � x + 5 �2 6
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Example 464. − 7 3 � 3 3 + 3 3
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9.4 Quadratics - Quadratic Formula
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x = Example 468. √ 30 ± 900 + 11
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9.4 Practice - Quadratic Formula So
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12x 2 − 17x + 6=0 Our Solution If
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9.5 Practice - Build Quadratics fro
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Example 479. a −2 − a −1 −
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12 = (x − 3) 2 or 13 =(x − 3) 2
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9.7 Quadratics - Rectangles Objecti
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Example 486. 2 Our Solution The len
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4(x 2 + 10x − 24) =0 Factor trino
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that require 24 square meters for p
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1(12x) 3 + 1(12x) x 1 1 5 + = 3 x 1
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6x +6(x − 1)=5x(x − 1) Reduce e
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16) A sink can be filled from the f
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These simultaneous product equation
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9.10 Quadratics - Revenue and Dista
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96n(n − 2) 96n(n − 2) +4n(n −
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upstream, the current will pull aga
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12) A pilot flying at a constant ra
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The above method to graph a parabol
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It is important to remember the gra
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Chapter 10 : Functions 10.1 Functio
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Example 504. Which of the following
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2x − 3 � 0 Set up an inequality
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Solve. 10.1 Practice - Function Not
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10.2 Functions - Operations on Func
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(x − 5)(x +1) (x − 5) Divide ou
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We can also evaluate a composition
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23) f(x) = x 2 − 5x g(x) = x + 5
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10.3 Functions - Inverse Functions
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(3x +6)(3 − 4x) 3 − 4x (4x+8)(3
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10.3 Practice - Inverse Functions S
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Example 530. 8 3x = 32 Rewrite 8 as
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10.4 Practice - Exponential Functio
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log3x =7 Identify base, 3, answer,
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10.5 Practice - Logarithmic Functio
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A = 25000(1.01625) 20 Evaluate expo
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The variable e is a constant simila
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j. All of the above compounded cont
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Using the diagram at right, find ea
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10.7 Practice - Trigonometric Funct
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21) 23) 25) 27) A B A 5 38 ◦ x x
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37) 39) A A 37.1 ◦ C x x C 13.1 4
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θ 17 From angle θ the given sides
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function to find the missing angle
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Find the measure of each angle indi
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27) 29) 14 A A θ C 15 7 C 14 θ B
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437
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7) 5 4 8) 4 3 9) 3 2 10) 8 3 11) 5
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54) 60v − 7 55) − 3x + 8x 2 56)
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45) 12 46) All real numbers 47) No
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15) wx 3 = 1458 16) h = 1.5 j 17) a
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1) B(4, − 3) C(1, 2) D( − 1, 4)
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1) y = 2x +5 2) y = − 6x +4 3) y
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23) x =4 24) y − 4 = 7 (x − 1)
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25) 5�x
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1) ( − 2, 4) 2) (2,4) 3) No solut
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27) 56, 144 28) 1.5, 3.5 29) 30 30)
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42) 1.2 × 10 6 5.4 1) 3 2) 7 3)
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5) 2x 3 + 4x 2 + x 2 6) 5p3 4 +4p2
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7) (b +8)(b + 4) 8) (b − 10)(b
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11) n(n − 1) 12) (5x +3)(x − 5)
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5) 3x2 2 6) 5p 2 7) 5m 8) 7 10 9) r
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7) 8) 5 24r 7x + 3y x 2 y 2 9) 15t+
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21) 0, 5 22) − 2, 5 3 23) 4,7 24)
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42) − 18xz 4x3yz3 � 4 8.3 1) 6
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16 3 √ + 4 5 √ 43 18) 19) 2 √
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8.8 1) 11 − 4i 2) − 4i 3) − 3
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√ √ 33) 4+i 39, 4 −i 39 34)
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9) ± 2, ± 4 10) 2,3, − 1 ±i 3
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1) 2) 3) 4) 5) 6) (-2,0) (0,-8) (-1
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11) 100 12) − 74 13) 1 5 14) 27 1
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10.5 1) 9 2 = 81 2) b −16 = a 3)
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21) 51.3 ◦ 22) 45 ◦ 23) 56.4
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