Beginning and Intermediate Algebra - Wallace Math Courses ...

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4.5 Systems of Equations - Value Problems Objective: Solve value problems by setting up a system of equations. One application of system of equations are known as value problems. Value problems are ones where each variable has a value attached to it. For example, if our variable is the number of nickles in a person’s pocket, those nickles would have a value of five cents each. We will use a table to help us set up and solve value problems. The basic structure of the table is shown below. Item 1 Item 2 Total Number Value Total The first column in the table is used for the number of things we have. Quite often, this will be our variables. The second column is used for the that value each item has. The third column is used for the total value which we calculate by multiplying the number by the value. For example, if we have 7 dimes, each with a value of 10 cents, the total value is 7 · 10 = 70 cents. The last row of the table is for totals. We only will use the third row (also marked total) for the totals that 158
are given to use. This means sometimes this row may have some blanks in it. Once the table is filled in we can easily make equations by adding each column, setting it equal to the total at the bottom of the column. This is shown in the following example. Example 186. In a child’s bank are 11 coins that have a value of S1.85. The coins are either quarters or dimes. How many coins each does child have? Number Value Total Quarter q 25 Dime d 10 Total Number Value Total Quarter q 25 25q Dime d 10 10d Total Number Value Total Quarter q 25 25q Dime d 10 10d Total 11 185 Using value table, use q for quarters, d for dimes Each quarter ′ s value is 25 cents, dime ′ s is 10 cents Multiply number by value to get totals We have 11 coins total. This is the number total. We have 1.85 for the final total, Write final total in cents (185) Because 25 and 10 are cents q +d=11 Firstandlastcolumnsareourequationsbyadding 25q + 10d = 185 Solve by either addition or substitution. − 10(q +d) =(11)( − 10) Using addition, multiply first equation by − 10 − 10q − 10d = − 110 − 10q − 10d = − 110 Add together equations 25q + 10d = 185 15q = 75 Divide both sides by 15 15 15 q = 5 We have our q, number of quarters is 5 (5) +d=11 Plug into one of original equations − 5 − 5 Subtract5from both sides d = 6 We have ourd, number of dimes is 6 159
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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
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Find each quotient. 37) − 2 ÷ 7
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the expression that is to be evalua
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Solve. 1) − 6 · 4( − 1) 3) 3+(
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It will be more common in our study
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0.4 Practice - Properties of Algebr
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Chapter 1 : Solving Linear Equation
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Addition Problems To solve equation
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8x = − 24 8 8 x = − 3 Division
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1.2 Linear Equations - Two-Step Equ
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− 1 − 1 Divide both sides by
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1.3 Solving Linear Equations - Gene
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Example 63. − 3x +9=6x − 27 Not
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4[ − 2]+9=−15 +8(2) Finish mult
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1.4 Solving Linear Equations - Frac
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Example 72. � � 3 5 4 x + = 3 2
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1.5 Solving Linear Equations - Form
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Example 79. A = πr 2 +πrs for s S
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1.5 Practice - Formulas Solve each
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|x| = 5 Absolute value can be posit
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Example 90. 7+|2x − 5| =4 Notice
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1.7 Solving Linear Equations - Vari
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Once we have found the constant of
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1.7 Practice - Variation Write the
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sure. If the pressure of a certain
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Fifteen more than three times a num
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When we started with our first, sec
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1.8 Practice - Number and Geometry
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30. The perimeter of a college bask
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Our equation comes from the future
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4x + 2=102 Solve the two − step e
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years ago, the bronze plaque was on
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1.10 Solving Linear Equations - Dis
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Rate Time Distance Bob r + 2 3 Fred
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2 =t Our solution fort, she catches
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Find the distance he rode. 8. A man
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The first plan is flying 25 mph slo
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2.1 Graphing - Points and Lines Obj
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E B D F G C A Our Solution The main
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x y − 3 − 4 0 − 2 3 0 Our com
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2.2 Graphing - Slope Objective: Fin
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When mathematicians began working w
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Example 131. Find the value of x be
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9) 10) Find the slope of the line t
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y = − 2 x + 3 Our Solution 3 Iden
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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
Page 119 and 120: than 4. If we have an expression su
Page 121 and 122: − 5 − 5 − 2x � 6 Divide bot
Page 123 and 124: Solve each inequality, graph each s
Page 125 and 126: As the graphs overlap, we take the
Page 127 and 128: 3.2 Practice - Compound Inequalitie
Page 129 and 130: edition of a college algebra text!
Page 131 and 132: It is important to remember as we a
Page 133 and 134: Chapter 4 : Systems of Equations 4.
Page 135 and 136: ested in the point that is a soluti
Page 137 and 138: that parallel lines have the same s
Page 139 and 140: 4.2 Systems of Equations - Substitu
Page 141 and 142: tion. This process is described and
Page 143 and 144: World View Note: French mathematici
Page 145 and 146: 31) 2x + y = 2 3x + 7y = 14 33) x +
Page 147 and 148: 6x +9y = 6 New second equation −
Page 149 and 150: Just as with graphing and substutio
Page 151 and 152: 4.4 Systems of Equations - Three Va
Page 153 and 154: just the ordered pair (x, y). In th
Page 155 and 156: 15x − 12y +9z = − 12 5x − 4y
Page 157: 23) 3x + 3y − 2z = 13 6x + 2y −
Page 161 and 162: − 0.5 − 0.5 c=17 We have c, num
Page 163 and 164: − 0.06x − 0.06y = − 240 Add e
Page 165 and 166: 14) A bank contains 27 coins in dim
Page 167 and 168: 4.6 Systems of Equations - Mixture
Page 169 and 170: Amount Part Total Start 40 3 120 Ad
Page 171 and 172: 4(30 −n)+2.5n=105 Substitute into
Page 173 and 174: 16) A certain grade of milk contain
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Page 177 and 178: 5.1 Polynomials - Exponent Properti
Page 179 and 180: A quicker method to arrive at the s
Page 181 and 182: Example 210. Example 211. 7a 3 (2a
Page 183 and 184: 5.2 Polynomials - Negative Exponent
Page 185 and 186: As we simplified our fraction we to
Page 187 and 188: 5.2 Practice - Negative Exponents S
Page 189 and 190: Example 223. Convert 3.21 × 10 5 t
Page 191 and 192: 5.3 Practice - Scientific Notation
Page 193 and 194: World View Note: Ada Lovelace in 18
Page 195 and 196: 21) (3+b 4 )+(7+2b + b 4 ) 22) (1+6
Page 197 and 198: Just as we distribute a monomial th
Page 199 and 200: Example 243. (2x − 5)(4x 2 − 7x
Page 201 and 202: 5.6 Polynomials - Multiply Special
Page 203 and 204: Be very careful when we are squarin
Page 205 and 206: 5.7 Polynomials - Divide Polynomial
Page 207 and 208: 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)
Page 489:
21) 51.3 ◦ 22) 45 ◦ 23) 56.4
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