Chapter 1

Determine whether each statement is true or false. If false, give a counterexample. A counterexample is a specific case that shows that a statement is false. Every integer is a whole number.

For Exercises \(58-63,\) define a variable, write an equation, and solve the problem. AGES Chun-Wei's mother is 8 more than twice his age. His father is three years older than his mother is. If the three family members have lived a total of 94 years, how old is each family member?

Solve each equation. Check your solutions. $$ |2 x+7|=15 $$

What is the complete solution to the equation \(|8-4 x|=40 ?\) F. \(x=8 ; x=12\) G. \(x=8 ; x=-12\) H. \(x=-8 ; x=-12\) J. \(x=-8 ; x=12\)

Determine whether each statement is true or false. If false, give a counterexample. A counterexample is a specific case that shows that a statement is false. Every real number is irrational.

For Exercises \(58-63,\) define a variable, write an equation, and solve the problem. School TRIP A Parent Teacher Organization has raised \(\$ 1800\) to help pay for a trip to an amusement park. They ask that there be one adult for every five students attending. Adult tickets are \(\$ 45\) and student tickets are \(\$ 30 .\) If the group wants to take 50 students, how much will each student need to pay so that adults agreeing to chaperone pay nothing?

Solve each equation. Check your solutions. \(|x-3|=17\)

Determine whether each statement is true or false. If false, give a counterexample. A counterexample is a specific case that shows that a statement is false. Every integer is a rational number.

For Exercises \(58-63,\) define a variable, write an equation, and solve the problem. Business A trucking company is hired to deliver 125 lamps for \(\$ 12\) each. The company agrees to pay \(\$ 45\) for each lamp that is broken during transport. If the trucking company needs to receive a minimum payment of \(\$ 1364\) for the shipment to cover their expenses, find the maximum number of lamps they can afford to break during the trip.

Solve each equation. \(14 y-3=25\)

Name the property illustrated by each statement. If \(3 x=10,\) then \(3 x+7=10+7\)

Solve each equation. Check your solutions. \(8|4 x-3|=64\)

REASONING Is the Distributive Property also true for division? In other words, does \(\frac{b+c}{a}=\frac{b}{a}+\frac{c}{a}, a \neq 0 ?\) If so, give an example and explain why it is true. If not true, give a counterexample.

Solve each equation. \(4.2 x+6.4=40\)

Name the property illustrated by each statement. If \(-5=4 y-8,\) then \(4 y-8=-5\)

Solve each equation. Check your solutions. \(|x+1|=x\)

RAILROADS For Exercises \(64-66,\) use the following information. The First Transcontinental Railroad was built by two companies. The Central Pacific began building eastward from Sacramento, California, while the Union Pacific built westward from Omaha, Nebraska. The two lines met at Promontory, Utah, in \(1869,\) approximately 6 years after construction began. The Central Pacific Company laid an average of 9.6 miles of track per month. Together the two companies laid a total of 1775 miles of track. Determine the average number of miles of track laid per month by the Union Pacific Company.

Solve each equation. \(7 w+2=3 w-6\)

Name the property illustrated by each statement. If \(-2 x-5=9\) and \(9=6 x+1,\) then \(-2 x-5=6 x+1\)

RAILROADS For Exercises \(64-66,\) use the following information. The First Transcontinental Railroad was built by two companies. The Central Pacific began building eastward from Sacramento, California, while the Union Pacific built westward from Omaha, Nebraska. The two lines met at Promontory, Utah, in \(1869,\) approximately 6 years after construction began. About how many miles of track did each company lay?

If \(a\) and \(b\) are natural numbers, then which of the following must also be a natural number? $$ \begin{array}{lll}{\text { I. } a-b} & {\text { II. } a b} & {\text { III. } \frac{a}{b}} \\ {\text { A I only }} & {\text { C III only }} \\ {\text { B II only }} & {\text { D I and II only }}\end{array} $$

Solve each equation. \(2(a-1)=8 a-6\)

Name the sets of numbers to which each number belongs. 31

Which equation is equivalent to \(4(9-3 x)=7-2(6-5 x) ?\) $$ \begin{array}{ll}{\mathbf{F}8 x=41} & {\mathbf{H} 22 x=41} \\ {\mathbf{G} 8 x=24} & {\mathbf{J} 22 x=24}\end{array} $$

Name the sets of numbers to which each number belongs. \(-4 . \overline{2}\)

Evaluate each expression. (lesson \(1-1 )\) $$ 9(4-3)^{5} $$

MONEY Allison is saving money to buy a video game system. In the first week, her savings were \(\$ 8\) less than \(\frac{2}{5}\) the price of the system. In the second week, she saved 50 cents more than \(\frac{1}{2}\) the price of the system. She was still \(\$ 37\) short. Find the price of the system.

Simplify each expression. $$ 6 a-2 b-3 a+9 b $$

Name the sets of numbers to which each number belongs. \(\sqrt{7}\)

Evaluate each expression. (lesson \(1-1 )\) $$ 5+9 \div 3(3)-8 $$

Simplify each expression. $$ -2(m-4 n)-3(5 n+6) $$

Jenny baby-sat for 5\(\frac{1}{2}\) hours on Friday night and 8 hours on Saturday. She charges \(\$ 4.25\) per hour. Use the Distributive Property to write two equivalent expressions that represent how much money Jenny earned.

Evaluate each expression if \(a=-5, b=0.25, c=\frac{1}{2},\) and \(d=4 .(\text { lesson } 1-1)\) $$ a+2 b-c $$

OPEN ENDED Write a two-step equation with a solution of \(-7 .\)

Find the value of each expression. $$ 6(5-8) \div 9+4 $$

Solve each equation. Check your solutions. \(|x|=7\)

Evaluate each expression if \(a=-5, b=0.25, c=\frac{1}{2},\) and \(d=4 .(\text { lesson } 1-1)\) $$ b+3(a+d)^{3} $$

REASONING Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. Dividing each side of an equation by the same expression produces an equivalent equation.

Find the value of each expression. $$ (3+7)^{2}-16 \div 2 $$

Solve each equation. Check your solutions. \(|x+5|=18\)

CHALLENGE Compare and contrast the Symmetric Property of Equality and the Commutative Property of Addition.

Find the value of each expression. $$ \frac{7(1-4)}{8-5} $$

Solve each equation. Check your solutions. \(|5 y-8|=12\)

PREREQUISITE SKILL Evaluate each expression if \(a=2, b=-\frac{3}{4},\) and \(c=1.8 .(\text { lesson } 1-1)\) $$ 8 b-5 $$

Solve each equation. Check your solutions. \(14=|2 x-36|\)

PREREQUISITE SKILL Evaluate each expression if \(a=2, b=-\frac{3}{4},\) and \(c=1.8 .(\text { lesson } 1-1)\) $$ \frac{2}{5} b+1 $$

ACT/SAT In triangle \(P Q R, \overline{Q S}\) and \(\overline{S R}\) are angle bisectors and angle \(P=74^{\circ} .\) How many degrees are there in angle \(Q S R ?\)

Solve each equation. Check your solutions. \(10=2|w+6|\)

REVIEW Which of the following best describes the graph of the equations below? $$ \begin{array}{l}{8 y=2 x+13} \\ {24 y=6 x+13}\end{array} $$ F The lines have the same \(y\) -intercept. G The lines have the same \(x\) -intercept. H The lines are perpendicular. J The lines are parallel.

PREREQUISITE SKILL Evaluate each expression if \(a=2, b=-\frac{3}{4},\) and \(c=1.8 .(\text { lesson } 1-1)\) $$ 1.5 c-7 $$