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Mathematics LibreTexts

Chapter 2 Review Exercises

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    30487
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    Chapter 2 Review Exercises

    Solve Equations using the Subtraction and Addition Properties of Equality

    Verify a Solution of an Equation

    In the following exercises, determine whether each number is a solution to the equation.

    Exercise \(\PageIndex{1}\)

    \(10 x-1=5 x ; x=\frac{1}{5}\)

    Exercise \(\PageIndex{2}\)

    \(w+2=\frac{5}{8} ; w=\frac{3}{8}\)

    Answer

    no

    Exercise \(\PageIndex{3}\)

    \(-12 n+5=8 n ; n=-\frac{5}{4}\)

    Exercise \(\PageIndex{4}\)

    \(6 a-3=-7 a, a=\frac{3}{13}\)

    Answer

    yes

    Solve Equations using the Subtraction and Addition Properties of Equality

    In the following exercises, solve each equation using the Subtraction Property of Equality.

    Exercise \(\PageIndex{5}\)

    \(x+7=19\)

    Exercise \(\PageIndex{6}\)

    \(y+2=-6\)

    Answer

    \(y=-8\)

    Exercise \(\PageIndex{7}\)

    \(a+\frac{1}{3}=\frac{5}{3}\)

    Exercise \(\PageIndex{8}\)

    \(n+3.6=5.1\)

    Answer

    \(n=1.5\)

    In the following exercises, solve each equation using the Addition Property of Equality.

    Exercise \(\PageIndex{9}\)

    \(u-7=10\)

    Exercise \(\PageIndex{10}\)

    \(x-9=-4\)

    Answer

    \(x=5\)

    Exercise \(\PageIndex{11}\)

    \(c-\frac{3}{11}=\frac{9}{11}\)

    Exercise \(\PageIndex{12}\)

    \(p-4.8=14\)

    Answer

    \(p=18.8\)

    In the following exercises, solve each equation.

    Exercise \(\PageIndex{13}\)

    \(n-12=32\)

    Exercise \(\PageIndex{14}\)

    \(y+16=-9\)

    Answer

    \(y=-25\)

    Exercise \(\PageIndex{15}\)

    \(f+\frac{2}{3}=4\)

    Exercise \(\PageIndex{16}\)

    \(d-3.9=8.2\)

    Answer

    \(d=12.1\)

    Solve Equations That Require Simplification

    In the following exercises, solve each equation.

    Exercise \(\PageIndex{17}\)

    \(y+8-15=-3\)

    Exercise \(\PageIndex{18}\)

    \(7 x+10-6 x+3=5\)

    Answer

    \(x=-8\)

    Exercise \(\PageIndex{19}\)

    \(6(n-1)-5 n=-14\)

    Exercise \(\PageIndex{20}\)

    \(8(3 p+5)-23(p-1)=35\)

    Answer

    \(p=-28\)

    Translate to an Equation and Solve

    In the following exercises, translate each English sentence into an algebraic equation and then solve it.

    Exercise \(\PageIndex{21}\)

    The sum of \(-6\) and \(m\) is 25

    Exercise \(\PageIndex{22}\)

    Four less than \(n\) is 13

    Answer

    \(n-4=13 ; n=17\)

    Translate and Solve Applications

    In the following exercises, translate into an algebraic equation and solve.

    Exercise \(\PageIndex{23}\)

    Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son?

    Exercise \(\PageIndex{24}\)

    Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh?

    Answer

    161 pounds

    Exercise \(\PageIndex{25}\)

    Peter paid $9.75 to go to the movies, which was $46.25 less than he paid to go to a concert. How much did he pay for the concert?

    Exercise \(\PageIndex{26}\)

    Elissa earned \(\$ 152.84\) this week, which was \(\$ 2 . .65\) more than she earned last week. How much did she earn last week?

    Answer

    \(\$ 131.19\)

    Solve Equations using the Division and Multiplication Properties of Equality

    Solve Equations Using the Division and Multiplication Properties of Equality

    In the following exercises, solve each equation using the division and multiplication properties of equality and check the solution.

    Exercise \(\PageIndex{27}\)

    \(8 x=72\)

    Exercise \(\PageIndex{28}\)

    \(13 a=-65\)

    Answer

    \(a=-5\)

    Exercise \(\PageIndex{29}\)

    \(0.25 p=5.25\)

    Exercise \(\PageIndex{30}\)

    \(-y=4\)

    Answer

    \(y=-4\)

    Exercise \(\PageIndex{31}\)

    \(\frac{n}{6}=18\)

    Exercise \(\PageIndex{32}\)

    \(\frac{y}{-10}=30\)

    Answer

    \(y=-300\)

    Exercise \(\PageIndex{33}\)

    \(36=\frac{3}{4} x\)

    Exercise \(\PageIndex{34}\)

    \(\frac{5}{8} u=\frac{15}{16}\)

    Answer

    \(u=\frac{3}{2}\)

    Exercise \(\PageIndex{35}\)

    \(-18 m=-72\)

    Exercise \(\PageIndex{36}\)

    \(\frac{c}{9}=36\)

    Answer

    \(c=324\)

    Exercise \(\PageIndex{37}\)

    \(0.45 x=6.75\)

    Exercise \(\PageIndex{38}\)

    \(\frac{11}{12}=\frac{2}{3} y\)

    Answer

    \(y=\frac{11}{8}\)

    Solve Equations That Require Simplification

    In the following exercises, solve each equation requiring simplification.

    Exercise \(\PageIndex{39}\)

    \(5 r-3 r+9 r=35-2\)

    Exercise \(\PageIndex{40}\)

    \(24 x+8 x-11 x=-7-14\)

    Answer

    \(x=-1\)

    Exercise \(\PageIndex{41}\)

    \(\frac{11}{12} n-\frac{5}{6} n=9-5\)

    Exercise \(\PageIndex{42}\)

    \(-9(d-2)-15=-24\)

    Answer

    \(d=3\)

    Translate to an Equation and Solve

    In the following exercises, translate to an equation and then solve.

    Exercise \(\PageIndex{43}\)

    143 is the product of \(-11\) and \(y\)

    Exercise \(\PageIndex{44}\)

    The quotient of \(b\) and and 9 is \(-27\)

    Answer

    \(\frac{b}{9}=-27 ; b=-243\)

    Exercise \(\PageIndex{45}\)

    The sum of q and one-fourth is one.

    Exercise \(\PageIndex{46}\)

    The difference of s and one-twelfth is one fourth.

    Answer

    \(s-\frac{1}{12}=\frac{1}{4} ; s=\frac{1}{3}\)

    Translate and Solve Applications

    In the following exercises, translate into an equation and solve.

    Exercise \(\PageIndex{47}\)

    Ray paid $21 for 12 tickets at the county fair. What was the price of each ticket?

    Exercise \(\PageIndex{48}\)

    Janet gets paid \(\$ 24\) per hour. She heard that this is \(\frac{3}{4}\) of what Adam is paid. How much is Adam paid per hour?

    Answer

    $32

    Solve Equations with Variables and Constants on Both Sides

    Solve an Equation with Constants on Both Sides

    In the following exercises, solve the following equations with constants on both sides.

    Exercise \(\PageIndex{49}\)

    \(8 p+7=47\)

    Exercise \(\PageIndex{50}\)

    \(10 w-5=65\)

    Answer

    \(w=7\)

    Exercise \(\PageIndex{51}\)

    \(3 x+19=-47\)

    Exercise \(\PageIndex{52}\)

    \(32=-4-9 n\)

    Answer

    \(n=-4\)

    Solve an Equation with Variables on Both Sides

    In the following exercises, solve the following equations with variables on both sides.

    Exercise \(\PageIndex{53}\)

    \(7 y=6 y-13\)

    Exercise \(\PageIndex{54}\)

    \(5 a+21=2 a\)

    Answer

    \(a=-7\)

    Exercise \(\PageIndex{55}\)

    \(k=-6 k-35\)

    Exercise \(\PageIndex{56}\)

    \(4 x-\frac{3}{8}=3 x\)

    Answer

    \(x=\frac{3}{8}\)

    Solve an Equation with Variables and Constants on Both Sides

    In the following exercises, solve the following equations with variables and constants on both sides.

    Exercise \(\PageIndex{57}\)

    \(12 x-9=3 x+45\)

    Exercise \(\PageIndex{58}\)

    \(5 n-20=-7 n-80\)

    Answer

    \(n=-5\)

    Exercise \(\PageIndex{59}\)

    \(4 u+16=-19-u\)

    Exercise \(\PageIndex{60}\)

    \(\frac{5}{8} c-4=\frac{3}{8} c+4\)

    Answer

    \(c=32\)

    Use a General Strategy for Solving Linear Equations

    Solve Equations Using the General Strategy for Solving Linear Equations

    In the following exercises, solve each linear equation.

    Exercise \(\PageIndex{61}\)

    \(6(x+6)=24\)

    Exercise \(\PageIndex{62}\)

    \(9(2 p-5)=72\)

    Answer

    \(p=\frac{13}{2}\)

    Exercise \(\PageIndex{63}\)

    \(-(s+4)=18\)

    Exercise \(\PageIndex{64}\)

    \(8+3(n-9)=17\)

    Answer

    \(n=12\)

    Exercise \(\PageIndex{65}\)

    \(23-3(y-7)=8\)

    Exercise \(\PageIndex{66}\)

    \(\frac{1}{3}(6 m+21)=m-7\)

    Answer

    \(m=-14\)

    Exercise \(\PageIndex{67}\)

    \(4(3.5 y+0.25)=365\)

    Exercise \(\PageIndex{68}\)

    \(0.25(q-8)=0.1(q+7)\)

    Answer

    \(q=18\)

    Exercise \(\PageIndex{69}\)

    \(8(r-2)=6(r+10)\)

    Exercise \(\PageIndex{70}\)

    \(\begin{array}{l}{5+7(2-5 x)=2(9 x+1)} \\ {-(13 x-57)}\end{array}\)

    Answer

    \(x=-1\)

    Exercise \(\PageIndex{71}\)

    \(\begin{array}{l}{(9 n+5)-(3 n-7)} \\ {=20-(4 n-2)}\end{array}\)

    Exercise \(\PageIndex{72}\)

    \(\begin{array}{l}{2[-16+5(8 k-6)]} \\ {=8(3-4 k)-32}\end{array}\)

    Answer

    \(k=\frac{3}{4}\)

    Classify Equations

    In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

    Exercise \(\PageIndex{73}\)

    \(\begin{array}{l}{17 y-3(4-2 y)=11(y-1)} \\ {+12 y-1}\end{array}\)

    Exercise \(\PageIndex{74}\)

    \(\begin{array}{l}{9 u+32=15(u-4)} \\ {-3(2 u+21)}\end{array}\)

    Answer

    contradiction; no solution

    Exercise \(\PageIndex{75}\)

    \(-8(7 m+4)=-6(8 m+9)\)

    Exercise \(\PageIndex{76}\)

    \(\begin{array}{l}{21(c-1)-19(c+1)} \\ {=2(c-20)}\end{array}\)

    Answer

    identity; all real numbers

    Solve Equations with Fractions and Decimals

    Solve Equations with Fraction Coefficients

    In the following exercises, solve each equation with fraction coefficients.

    Exercise \(\PageIndex{77}\)

    \(\frac{2}{5} n-\frac{1}{10}=\frac{7}{10}\)

    Exercise \(\PageIndex{78}\)

    \(\frac{1}{3} x+\frac{1}{5} x=8\)

    Answer

    \(x=15\)

    Exercise \(\PageIndex{79}\)

    \(\frac{3}{4} a-\frac{1}{3}=\frac{1}{2} a-\frac{5}{6}\)

    Exercise \(\PageIndex{80}\)

    \(\frac{1}{2}(k-3)=\frac{1}{3}(k+16)\)

    Answer

    \(k=41\)

    Exercise \(\PageIndex{81}\)

    \(\frac{3 x-2}{5}=\frac{3 x+4}{8}\)

    Exercise \(\PageIndex{82}\)

    \(\frac{5 y-1}{3}+4=\frac{-8 y+4}{6}\)

    Answer

    \(y=-1\)

    Solve Equations with Decimal Coefficients

    In the following exercises, solve each equation with decimal coefficients.

    Exercise \(\PageIndex{83}\)

    \(0.8 x-0.3=0.7 x+0.2\)

    Exercise \(\PageIndex{84}\)

    \(0.36 u+2.55=0.41 u+6.8\)

    Answer

    \(u=-85\)

    Exercise \(\PageIndex{85}\)

    \(0.6 p-1.9=0.78 p+1.7\)

    Exercise \(\PageIndex{86}\)

    \(0.6 p-1.9=0.78 p+1.7\)

    Answer

    \(d=-20\)

    Solve a Formula for a Specific Variable

    Use the Distance, Rate, and Time Formula

    In the following exercises, solve.

    Exercise \(\PageIndex{87}\)

    Natalie drove for 7\(\frac{1}{2}\) hours at 60 miles per hour. How much distance did she travel?

    Exercise \(\PageIndex{88}\)

    Mallory is taking the bus from St. Louis to Chicago. The distance is 300 miles and the bus travels at a steady rate of 60 miles per hour. How long will the bus ride be?

    Answer

    5 hours

    Exercise \(\PageIndex{89}\)

    Aaron’s friend drove him from Buffalo to Cleveland. The distance is 187 miles and the trip took 2.75 hours. How fast was Aaron’s friend driving?

    Exercise \(\PageIndex{90}\)

    Link rode his bike at a steady rate of 15 miles per hour for 2\(\frac{1}{2}\) hours. How much distance did he travel?

    Answer

    37.5 miles

    Solve a Formula for a Specific Variable

    In the following exercises, solve.

    Exercise \(\PageIndex{91}\)

    Use the formula. d=rt to solve for t

    1. when d=510 and r=60
    2. in general

    Exercise \(\PageIndex{92}\)

    Use the formula. d=rt to solve for r

    1. when when d=451 and t=5.5
    2. in general
    Answer
    1. r=82mph
    2. \(r=\frac{D}{t}\)

    Exercise \(\PageIndex{93}\)

    Use the formula \(A=\frac{1}{2} b h\) to solve for b

    1. when A=390 and h=26
    2. in general

    Exercise \(\PageIndex{94}\)

    Use the formula \(A=\frac{1}{2} b h\) to solve for b

    1. when A=153 and b=18
    2. in general
    Answer
    1. \(h=17\)
    2. \( h=\frac{2 A}{b}\)

    Exercise \(\PageIndex{95}\)

    Use the formula I=Prt to solve for the principal, P for

    1. I=$2,501,r=4.1%, t=5 years
    2. in general

    Exercise \(\PageIndex{96}\)

    Solve the formula 4x+3y=6 for y

    1. when x=−2
    2. in general
    Answer

    ⓐ \(y=\frac{14}{3}\) ⓑ \( y=\frac{6-4 x}{3}\)

    Exercise \(\PageIndex{97}\)

    Solve \(180=a+b+c\) for \(c\)

    Exercise \(\PageIndex{98}\)

    Solve the formula \(V=L W H\) for \(H\)

    Answer

    \(H=\frac{V}{L W}\)

    Solve Linear Inequalities

    Graph Inequalities on the Number Line

    In the following exercises, graph each inequality on the number line.

    Exercise \(\PageIndex{99}\)

    1. \(x\leq 4\)
    2. x>−2
    3. x<1

    Exercise \(\PageIndex{100}\)

    1. x>0
    2. x<−3
    3. \(x\geq −1\)
    Answer
    1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 0 is graphed on the number line, with an open parenthesis at x equals 0, and a dark line extending to the right of the parenthesis.
    2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 3 is graphed on the number line, with an open parenthesis at x equals negative 3, and a dark line extending to the left of the parenthesis.
    3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a dark line extending to the right of the bracket.

    In the following exercises, graph each inequality on the number line and write in interval notation.

    Exercise \(\PageIndex{101}\)

    1. \(x<-1\)
    2. \(x \geq-2.5\)
    3. \(x \leq \frac{5}{4}\)

    Exercise \(\PageIndex{102}\)

    1. \(x>2\)
    2. \(x \leq-1.5\)
    3. \(x \geq \frac{5}{3}\)
    Answer
    1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, 2 comma infinity, parenthesis.
    2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1.5 is graphed on the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 1.5, bracket.
    3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 5/3 is graphed on the number line, with an open bracket at x equals 5/3, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 5/3 comma infinity, parenthesis.

    Solve Inequalities using the Subtraction and Addition Properties of Inequality

    In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    Exercise \(\PageIndex{103}\)

    \(n-12 \leq 23\)

    Exercise \(\PageIndex{104}\)

    \(m+14 \leq 56\)

    Answer

    At the top of this figure is the solution to the inequality: m is less than or equal to 42. Below this is a number line ranging from 40 to 44 with tick marks for each integer. The inequality m is less than or equal to 42 is graphed on the number line, with an open bracket at m equals 42, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 42, bracket

    Exercise \(\PageIndex{105}\)

    \(a+\frac{2}{3} \geq \frac{7}{12}\)

    Exercise \(\PageIndex{106}\)

    \(b-\frac{7}{8} \geq-\frac{1}{2}\)

    Answer

    At the top of this figure is the solution to the inequality: b is greater than or equal to 3/8. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. The inequality b is greater than or equal to 3/8 is graphed on the number line, with an open bracket at b equals 3/8 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 3/8 comma infinity, bracket

    Solve Inequalities using the Division and Multiplication Properties of Inequality

    In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    Exercise \(\PageIndex{107}\)

    \(9 x>54\)

    Exercise \(\PageIndex{108}\)

    \(-12 d \leq 108\)

    Answer

    At the top of this figure is the solution to the inequality: d is greater than or equal to negative 9. Below this is a number line ranging from negative 11 to negative 7 with tick marks for each integer. The inequality d is greater than or equal to negative 9 is graphed on the number line, with an open bracket at d equals negative 9, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 9 comma infinity, parenthesis.

    Exercise \(\PageIndex{109}\)

    \(\frac{5}{2} j<-60\)

    Exercise \(\PageIndex{110}\)

    \(\frac{q}{-2} \geq-24\)

    Answer

    At the top of this figure is the solution to the inequality: q is less than or equal to 48. Below this is a number line ranging from 46 to 50 with tick marks for each integer. The inequality q is less than or equal to 48 is graphed on the number line, with an open bracket at q equals 48, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 48, bracket.

    Solve Inequalities That Require Simplification

    In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    Exercise \(\PageIndex{111}\)

    \(6 p>15 p-30\)

    Exercise \(\PageIndex{112}\)

    \(9 h-7(h-1) \leq 4 h-23\)

    Answer

    At the top of this figure is the solution to the inequality: h is greater than or equal to 15. Below this is a number line ranging from 13 to 17 with tick marks for each integer. The inequality h is greater than or equal to 15 is graphed on the number line, with an open bracket at h equals 15, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 15 comma infinity, parenthesis.

    Exercise \(\PageIndex{113}\)

    \(5 n-15(4-n)<10(n-6)+10 n\)

    Exercise \(\PageIndex{114}\)

    \(\frac{3}{8} a-\frac{1}{12} a>\frac{5}{12} a+\frac{3}{4}\)

    Answer

    At the top of this figure is the solution to the inequality: a is less than negative 6. Below this is a number line ranging from negative 8 to negative 4 with tick marks for each integer. The inequality a is less than negative 6 is graphed on the number line, with an open parenthesis at a equals negative 6, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 6, parenthesis.

    Translate to an Inequality and Solve

    In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line.

    Exercise \(\PageIndex{115}\)

    Five more than z is at most 19.

    Exercise \(\PageIndex{116}\)

    Three less than c is at least 360.

    Answer

    At the top of this figure is the inequality c minus 3 is greater than or equal to 360. To the right of this is the solution to the inequality: c is greater than or equal to 363. To the right of the solution is the solution written in interval notation: bracket, 363 comma infinity, parenthesis. Below all of this is a number line ranging from 361 to 365 with tick marks for each integer. The inequality c is greater than or equal to 363 is graphed on the number line, with an open bracket at c equals 363, and a dark line extending to the right of the bracket.

    Exercise \(\PageIndex{117}\)

    Nine times n exceeds 42.

    Exercise \(\PageIndex{118}\)

    Negative two times a is no more than 8.

    Answer

    At the top of this figure is the inequality negative 2a is less than or equal to 8. To the right of this is the solution to the inequality: a is greater than or equal to negative 4. To the right of the solution is the solution written in interval notation: bracket, negative 4 comma infinity, parenthesis. Below all of this is a number line ranging from negative 6 to negative 2 with tick marks for each integer. The inequality a is greater than or equal to negative 4 is graphed on the number line, with an open bracket at a equals negative 4, and a dark line extending to the right of the bracket.

    Everyday Math

    Exercise \(\PageIndex{119}\)

    Describe how you have used two topics from this chapter in your life outside of your math class during the past month.

    Chapter 2 Practice Test

    Exercise \(\PageIndex{1}\)

    Determine whether each number is a solution to the equation \(6 x-3=x+20\)

    1. 5
    2. \(\frac{23}{5}\)
    Answer
    1. no
    2. yes

    In the following exercises, solve each equation.

    Exercise \(\PageIndex{2}\)

    \(n-\frac{2}{3}=\frac{1}{4}\)

    Exercise \(\PageIndex{3}\)

    \(\frac{9}{2} c=144\)

    Answer

    c=32

    Exercise \(\PageIndex{4}\)

    \(4 y-8=16\)

    Exercise \(\PageIndex{5}\)

    \(-8 x-15+9 x-1=-21\)

    Answer

    \(x=-5\)

    Exercise \(\PageIndex{6}\)

    \(-15 a=120\)

    Exercise \(\PageIndex{7}\)

    \(\frac{2}{3} x=6\)

    Answer

    \(x=9\)

    Exercise \(\PageIndex{8}\)

    \(x-3.8=8.2\)

    Exercise \(\PageIndex{9}\)

    \(10 y=-5 y-60\)

    Answer

    \(y=-4\)

    Exercise \(\PageIndex{10}\)

    \(8 n-2=6 n-12\)

    Exercise \(\PageIndex{11}\)

    \(9 m-2-4 m-m=42-8\)

    Answer

    \(m=9\)

    Exercise \(\PageIndex{12}\)

    \(-5(2 x-1)=45\)

    Exercise \(\PageIndex{13}\)

    \(-(d-9)=23\)

    Answer

    \(d=-14\)

    Exercise \(\PageIndex{14}\)

    \(\frac{1}{4}(12 m-28)=6-2(3 m-1)\)

    Exercise \(\PageIndex{15}\)

    \(2(6 x-5)-8=-22\)

    Answer

    \(x=-\frac{1}{3}\)

    Exercise \(\PageIndex{16}\)

    \(8(3 a-5)-7(4 a-3)=20-3 a\)

    Exercise \(\PageIndex{17}\)

    \(\frac{1}{4} p-\frac{1}{3}=\frac{1}{2}\)

    Answer

    \(p=\frac{10}{3}\)

    Exercise \(\PageIndex{18}\)

    \(0.1 d+0.25(d+8)=4.1\)

    Exercise \(\PageIndex{19}\)

    \(14 n-3(4 n+5)=-9+2(n-8)\)

    Answer

    contradiction; no solution

    Exercise \(\PageIndex{20}\)

    \(9(3 u-2)-4[6-8(u-1)]=3(u-2)\)

    Exercise \(\PageIndex{21}\)

    Solve the formula x−2y=5 for y

    1. when x=−3
    2. in general
    Answer
    1. y=4
    2. \(y=\frac{5-x}{2}\)

    In the following exercises, graph on the number line and write in interval notation.

    Exercise \(\PageIndex{22}\)

    \(x \geq-3.5\)

    Exercise \(\PageIndex{23}\)

    \(x<\frac{11}{4}\)

    Answer

    This figure is a number line ranging from 1 to 5 with tick marks for each integer. The inequality x is less than 11/4 is graphed on the number line, with an open parenthesis at x equals 11/4, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/4, parenthesis.

    In the following exercises,, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    Exercise \(\PageIndex{24}\)

    \(8 k \geq 5 k-120\)

    Exercise \(\PageIndex{25}\)

    \(3 c-10(c-2)<5 c+16\)

    Answer

    This figure is a number line ranging from negative 2 to 3 with tick marks for each integer. The inequality c is greater than 1/3 is graphed on the number line, with an open parenthesis at c equals 1/3, and a dark line extending to the right of the parenthesis. Below the number line is the solution: c is greater than 1/3. To the right of the solution is the solution written in interval notation: parenthesis, 1/3 comma infinity, parenthesis

    In the following exercises, translate to an equation or inequality and solve.

    Exercise \(\PageIndex{26}\)

    4 less than twice x is 16.

    Exercise \(\PageIndex{27}\)

    Fifteen more than n is at least 48.

    Answer

    \(n+15 \geq 48 ; n \geq 33\)

    Exercise \(\PageIndex{28}\)

    Samuel paid $25.82 for gas this week, which was $3.47 less than he paid last week. How much had he paid last week?

    Exercise \(\PageIndex{29}\)

    Jenna bought a coat on sale for \(\$ 120,\) which was \(\frac{2}{3}\) of the original price. What was the original price of the coat?

    Answer

    \(120=\frac{2}{3} p ;\) The original price was \(\$ 180\)

    Exercise \(\PageIndex{30}\)

    Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7\(\frac{2}{3}\) hours, what was the speed of the bus?

     

    Review for 2.7 Ratio and Proportions and Similar Triangles

    Solve Proportions

    In the following exercises, solve.

    Exercise \(\PageIndex{74}\)

    \(\dfrac{x}{4}=\dfrac{3}{5}\)

    Answer

    \(\dfrac{12}{5}\)

    Exercise \(\PageIndex{75}\)

    \(\dfrac{3}{y}=\dfrac{9}{5}\)

    Exercise \(\PageIndex{76}\)

    \(\dfrac{s}{s+20}=\dfrac{3}{7}\)

    Answer

    \(15\)

    Exercise \(\PageIndex{77}\)

    \(\dfrac{t−3}{5}=\dfrac{t+2}{9}\)

    ​​​​​​​In the following exercises, solve using proportions.

    Exercise \(\PageIndex{78}\)

    Rachael had a \(21\) ounce strawberry shake that has \(739\) calories. How many calories are there in a \(32\) ounce shake?

    Answer

    \(1161\) calories

    Exercise \(\PageIndex{79}\)

    Leo went to Mexico over Christmas break and changed \($525\) dollars into Mexican pesos. At that time, the exchange rate had \($1\) US is equal to \(16.25\) Mexican pesos. How many Mexican pesos did he get for his trip?

    ​​​​​​​Solve Similar Figure Applications

    In the following exercises, solve.

    Exercise \(\PageIndex{80}\)

    \(∆ABC\) is similar to \(∆XYZ\). The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.

    This image shows two triangles. The large triangle is labeled A B C. The length from A to B is labeled 8. The length from B to C is labeled 7. The length from C to A is labeled b. The smaller triangle is triangle x y z. The length from x to y is labeled 2 and two-thirds. The length from y to z is labeled x. The length from x to z is labeled 3.

    Answer

    \(b=9\); \(x=2\dfrac{1}{3}\)

    Exercise \(\PageIndex{81}\)

    On a map of Europe, Paris, Rome, and Vienna form a triangle whose sides are shown in the figure below. If the actual distance from Rome to Vienna is \(700\) miles, find the distance from

    1. a. Paris to Rome
    2. b. Paris to Vienna

    This is an image of a triangle. Clockwise beginning at the top, each vertex is labeled. The top vertex is labeled “Paris”, the next vertex is labeled “Vienna”, and the next vertex is labeled “Rome”. The distance from Paris to Vienna is 7.7 centimeters. The distance from Vienna to Rome is 7 centimeters. The distance from Rome to Paris is 8.9 centimeters.

    Exercise \(\PageIndex{82}\)

    Tony is \(5.75\) feet tall. Late one afternoon, his shadow was \(8\) feet long. At the same time, the shadow of a nearby tree was \(32\) feet long. Find the height of the tree.

    Answer

    \(23\) feet

    Exercise \(\PageIndex{83}\)

    The height of a lighthouse in Pensacola, Florida is \(150\) feet. Standing next to the statue, \(5.5\) foot tall Natalie cast a \(1.1\) foot shadow How long would the shadow of the lighthouse be?

     

    Review for 2.9 Compound Inequalities 

    Solve Compound Inequalities with “and”

    In each of the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.

    98. \(x\leq 5\) and \(x>−3\)

    Answer

    The solution is negative 3 is less than x which is less than or equal to 5. The number line shows an open circle at negative 3 and a closed circle at 5. The interval notation is negative 3 to 5 within a parenthesis and a bracket.

    99. \(4x−2\leq 4\) and \(7x−1>−8\)

    100. \(5(3x−2)\leq 5\) and \(4(x+2)<3\)

    Answer

    The solution is negative x is less than negative five-fourths. The number line shows an open circle at negative five-fourths with shading to its left. The interval notation is negative infinity to negative five-fourths within parentheses.

    101. \(34(x−8)\leq 3\) and \(15(x−5)\leq 3\)

    102. \(34x−5\geq −2\) and \(−3(x+1)\geq 6\)

    Answer

    The solution is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation

    103. \(−5\leq 4x−1<7\)

    Solve Compound Inequalities with “or”

    In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

    104. \(5−2x\leq −1\) or \(6+3x\leq 4\)

    Answer

    The solution is x is less than negative two-thirds or x is greater than or equal to 3. The number line shows a closed circle at negative two-thirds with shading to its left and a closed circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative two-thirds within a parenthesis and a bracket and 3 to infinity within a bracket and a parenthesis.

    105. \(3(2x−3)<−5\) or \(4x−1>3\)

    106. \(34x−2>4\) or \(4(2−x)>0\)

    Answer

    The solution is x is less than 2 or x is greater than 8. The number line shows an open circle at 2 with shading to its left and an open circle at 8 with shading to its right. The interval notation is the union of negative infinity to 8 within parentheses and 8 to infinity within parentheses.

    107. \(2(x+3)\geq 0\) or \(3(x+4)\leq 6\)

    108. \(12x−3\leq 4\) or \(13(x−6)\geq −2\)

    Answer

    The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.

    Solve Applications with Compound Inequalities

    In the following exercises, solve.

    109. Liam is playing a number game with his sister Audry. Liam is thinking of a number and wants Audry to guess it. Five more than three times her number is between 2 and 32. Write a compound inequality that shows the range of numbers that Liam might be thinking of.

    110. Elouise is creating a rectangular garden in her back yard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.

    Answer

    \(6\leq w\leq 12\)