Chapter 2 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

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Worked example

conditional equation

An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.

contradiction

An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution.

identity

An equation that is true for any value of the variable is called an identity. The solution of an identity is all real numbers.

solution of an equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

##

Key Concepts

• To Determine Whether a Number is a Solution to an Equation Step 1. Substitute the number in for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting statement is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

• Addition Property of Equality For any numbers a, b, and c, if $a=b$, then $a+c=b+c$.

• Subtraction Property of Equality For any numbers a, b, and c, if $a=b$, then $a-c=b-c$.

• To Translate a Sentence to an Equation Step 1. Locate the “equals” word(s). Translate to an equal sign (=). Step 2. Translate the words to the left of the “equals” word(s) into an algebraic expression. Step 3. Translate the words to the right of the “equals” word(s) into an algebraic expression.

• To Solve an Application Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with the important information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• The Division Property of Equality—For any numbers a, b, and c, and $c\ne{}0$, if $a=b$, then $\frac{a}{c}=\frac{b}{c}$. When you divide both sides of an equation by any non-zero number, you still have equality.

• The Multiplication Property of Equality—For any numbers a, b, and c, if $a=b$, then $ac=bc$. If you multiply both sides of an equation by the same number, you still have equality.

• Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation Step 1. Choose which side will be the “variable” side—the other side will be the “constant” side. Step 2. Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality. Step 3. Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality. Step 4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality. Step 5. Check the solution by substituting it into the original equation.

• General Strategy for Solving Linear Equations Step 1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms. Step 2. Collect all the variable terms on one side of the equation. Use the Addition or Subtraction Property of Equality. Step 3. Collect all the constant terms on the other side of the equation. Use the Addition or Subtraction Property of Equality. Step 4. Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation. Step 5. Check the solution. Substitute the solution into the original equation.

• Strategy to Solve an Equation with Fraction Coefficients Step 1. Find the least common denominator of all the fractions in the equation. Step 2. Multiply both sides of the equation by that LCD. This clears the fractions. Step 3. Solve using the General Strategy for Solving Linear Equations.

• To Solve an Application (with a formula) Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. Write the appropriate formula for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Distance, Rate and Time For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula: $d=rt$ where d = distance, r = rate, t = time.

• To solve a formula for a specific variable means to get that variable by itself with a coefficient of 1 on one side of the equation and all other variables and constants on the other side.

• Subtraction Property of Inequality For any numbers a, b, and c, if $a<b$ then $a-c<b-c$ and if $a>b$ then $a-c>b-c.$

• Addition Property of Inequality For any numbers a, b, and c, if $a<b$ then $a+c<b+c$ and if $a>b$ then $a+c>b+c.$

• Division and Multiplication Properties of Inequality For any numbers a, b, and c, if $a<b$ and $c>0$, then $\frac{a}{c}<\frac{b}{c}$ and $ac>bc$. if $a>b$ and $c>0$, then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$. if $a<b$ and $c<0$, then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$. if $a>b$ and $c<0$, then $\frac{a}{c}<\frac{b}{c}$ and $ac<bc$.

• When we divide or multiply an inequality by a: positive number, the inequality stays the same. negative number, the inequality reverses.

##

Review Exercises

Verify a Solution of an Equation

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

Solve Equations using the Subtraction and Addition Properties of Equality

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

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

In the following exercises, solve each equation.

Solve Equations That Require Simplification

In the following exercises, solve each equation.

Translate to an Equation and Solve

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

Translate and Solve Applications

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

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.

Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

Translate to an Equation and Solve

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

Translate and Solve Applications

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

Solve an Equation with Constants on Both Sides

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

Solve an Equation with Variables on Both Sides

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

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.

Solve Equations Using the General Strategy for Solving Linear Equations

In the following exercises, solve each linear equation.

Classify Equations

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

Solve Equations with Fraction Coefficients

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

Solve Equations with Decimal Coefficients

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

Use the Distance, Rate, and Time Formula

In the following exercises, solve.

Solve a Formula for a Specific Variable

In the following exercises, solve.

Graph Inequalities on the Number Line

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

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

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.

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.

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.

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.

##

In the following exercises, solve each equation.

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

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

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

Key Terms

conditional equation

An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.

contradiction

An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution.

identity

An equation that is true for any value of the variable is called an identity. The solution of an identity is all real numbers.

solution of an equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

##

Key Concepts

• To Determine Whether a Number is a Solution to an Equation Step 1. Substitute the number in for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting statement is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

• Addition Property of Equality For any numbers a, b, and c, if $a=b$, then $a+c=b+c$.

• Subtraction Property of Equality For any numbers a, b, and c, if $a=b$, then $a-c=b-c$.

• To Translate a Sentence to an Equation Step 1. Locate the “equals” word(s). Translate to an equal sign (=). Step 2. Translate the words to the left of the “equals” word(s) into an algebraic expression. Step 3. Translate the words to the right of the “equals” word(s) into an algebraic expression.

• To Solve an Application Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with the important information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• The Division Property of Equality—For any numbers a, b, and c, and $c\ne{}0$, if $a=b$, then $\frac{a}{c}=\frac{b}{c}$. When you divide both sides of an equation by any non-zero number, you still have equality.

• The Multiplication Property of Equality—For any numbers a, b, and c, if $a=b$, then $ac=bc$. If you multiply both sides of an equation by the same number, you still have equality.

• Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation Step 1. Choose which side will be the “variable” side—the other side will be the “constant” side. Step 2. Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality. Step 3. Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality. Step 4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality. Step 5. Check the solution by substituting it into the original equation.

• General Strategy for Solving Linear Equations Step 1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms. Step 2. Collect all the variable terms on one side of the equation. Use the Addition or Subtraction Property of Equality. Step 3. Collect all the constant terms on the other side of the equation. Use the Addition or Subtraction Property of Equality. Step 4. Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation. Step 5. Check the solution. Substitute the solution into the original equation.

• Strategy to Solve an Equation with Fraction Coefficients Step 1. Find the least common denominator of all the fractions in the equation. Step 2. Multiply both sides of the equation by that LCD. This clears the fractions. Step 3. Solve using the General Strategy for Solving Linear Equations.

• To Solve an Application (with a formula) Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. Write the appropriate formula for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Distance, Rate and Time For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula: $d=rt$ where d = distance, r = rate, t = time.

• To solve a formula for a specific variable means to get that variable by itself with a coefficient of 1 on one side of the equation and all other variables and constants on the other side.

• Subtraction Property of Inequality For any numbers a, b, and c, if $a<b$ then $a-c<b-c$ and if $a>b$ then $a-c>b-c.$

• Addition Property of Inequality For any numbers a, b, and c, if $a<b$ then $a+c<b+c$ and if $a>b$ then $a+c>b+c.$

• Division and Multiplication Properties of Inequality For any numbers a, b, and c, if $a<b$ and $c>0$, then $\frac{a}{c}<\frac{b}{c}$ and $ac>bc$. if $a>b$ and $c>0$, then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$. if $a<b$ and $c<0$, then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$. if $a>b$ and $c<0$, then $\frac{a}{c}<\frac{b}{c}$ and $ac<bc$.

• When we divide or multiply an inequality by a: positive number, the inequality stays the same. negative number, the inequality reverses.

##

Review Exercises

Verify a Solution of an Equation

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

Solve Equations using the Subtraction and Addition Properties of Equality

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

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

In the following exercises, solve each equation.

Solve Equations That Require Simplification

In the following exercises, solve each equation.

Translate to an Equation and Solve

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

Translate and Solve Applications

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

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.

Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

Translate to an Equation and Solve

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

Translate and Solve Applications

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

Solve an Equation with Constants on Both Sides

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

Solve an Equation with Variables on Both Sides

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

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.

Solve Equations Using the General Strategy for Solving Linear Equations

In the following exercises, solve each linear equation.

Classify Equations

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

Solve Equations with Fraction Coefficients

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

Solve Equations with Decimal Coefficients

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

Use the Distance, Rate, and Time Formula

In the following exercises, solve.

Solve a Formula for a Specific Variable

In the following exercises, solve.

Graph Inequalities on the Number Line

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

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

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.

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.

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.

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.

##

In the following exercises, solve each equation.

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

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

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

1. $w+2=\frac{5}{8};w=\frac{3}{8}$

2. $6a-3=-7a,a=\frac{3}{13}$

3. $y+2=-6$

4. $n+3.6=5.1$

5. $x-9=-4$

1. $p-4.8=14$

2. $y+16=-9$

3. $d-3.9=8.2$

4. $7x+10-6x+3=5$

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

1. Four less than $n$ is 13.

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

3. Elissa earned $152.84 this week, which was$21.65 more than she earned last week. How much did she earn last week?

4. $13a=-65$

5. $-y=4$

1. $\frac{y}{-10}=30$

2. $\frac{5}{8}u=\frac{15}{16}$

3. $\frac{c}{9}=36$

4. $\frac{11}{12}=\frac{2}{3}y$

5. $24x+8x-11x=-7-14$

1. $-9(d-2)-15=-24$

2. The quotient of b and 9 is $-27$.

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

4. 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?

5. $10w-5=65$

1. $32=-4-9n$

2. $5a+21=2a$

3. $4x-\frac{3}{8}=3x$

4. $5n-20=-7n-80$

5. $\frac{5}{8}c-4=\frac{3}{8}c+4$

1. $9(2p-5)=72$

2. $8+3(n-9)=17$

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

4. $0.25(q-8)=0.1(q+7)$

5. $5+7(2-5x)=2(9x+1)$$-(13x-57)$

1. $2[-16+5(8k-6)]$$=8(3-4k)-32$

2. $9u+32=15(u-4)$$-3(2u+21)$

3. $21(c-1)-19(c+1)$$=2(c-20)$

4. $\frac{1}{3}x+\frac{1}{5}x=8$

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

1. $\frac{5y-1}{3}+4=\frac{-8y+4}{6}$

2. $0.36u+2.55=0.41u+6.8$

3. $0.7y+2.5=0.95y-9.25$

4. 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?

5. 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?

1. Use the formula. $d=rt$ to solve for $r$ⓐ when when $d=451$ and $t=5.5$ⓑ in general

2. Use the formula $A=\frac{1}{2}bh$ to solve for $h$ⓐ when $A=153$ and $b=18$ⓑ in general

3. Solve the formula $4x+3y=6$ for yⓐ when $x=-2$ⓑ in general

4. Solve the formula $V=LWH$ for $H$.

5. ⓐ $x>0$ⓑ $x<-3$ⓒ $x\ge-1$

1. ⓐ $x>2$ⓑ $x\le-1.5$ⓒ $x\ge\frac{5}{3}$

2. $m+14\le{}56$

3. $b-\frac{7}{8}\ge-\frac{1}{2}$

4. $-12d\le{}108$

5. $\frac{q}{-2}\ge-24$

1. $9h-7(h-1)\le{}4h-23$

2. $\frac{3}{8}a-\frac{1}{12}a>\frac{5}{12}a+\frac{3}{4}$

3. Three less than c is at least 360.

4. Negative two times a is no more than 8.

5. Determine whether each number is a solution to the equation $6x-3=x+20$.

ⓐ 5ⓑ $\frac{23}{5}$

1. $\frac{9}{2}c=144$

2. $-8x-15+9x-1=-21$

3. $\frac{2}{3}x=6$

4. $10y=-5y-60$

5. $9m-2-4m-m=42-8$

1. $-(d-9)=23$

2. $2(6x-5)-8=-22$

3. $\frac{1}{4}p-\frac{1}{3}=\frac{1}{2}$

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

5. Solve the formula $x-2y=5$ for yⓐ when $x=-3$ⓑ in general

1. $x<\frac{11}{4}$

2. $3c-10(c-2)<5c+16$

3. Fifteen more than n is at least 48.

4. 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?