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.

##

Worked example

compound inequality

A compound inequality is made up of two inequalities connected by the word “and” or the word “or.”

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.

linear equation

a

b

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

• How 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 equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

• How to Solve Linear Equations Using a General Strategy 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 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 to make sure the result is a true statement.

• How to Solve Equations with Fraction or Decimal Coefficients Step 1. Find the least common denominator (LCD) of all the fractions and decimals (in fraction form) in the equation. Step 2. Multiply both sides of the equation by that LCD. This clears the fractions and decimals. Step 3. Solve using the General Strategy for Solving Linear Equations.

• How To Use a Problem Solving Strategy for Word Problems Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what you are looking for. Step 3. Name what you 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 all the important information. Then, translate the English sentence into an algebra equation. Step 5. Solve the equation using proper algebra techniques. Step 6. Check the answer in the problem to make sure it makes sense. Step 7. Answer the question with a complete sentence.

• How To Find Percent Change Step 1. Find the amount of change$\mathrm{change}=\mathrm{new amount}-\mathrm{original amount}$ Step 2. Find what percent the amount of change is of the original amount.$\mathrm{change is what percent of the original amount?}$

• Discount$\begin{array}{rll} \mathrm{amount of discount} & = & \mathrm{discount rate}\cdot\mathrm{original price} \\ \mathrm{sale price} & = & \mathrm{original amount}-\mathrm{discount} \\ \end{array}$

• Mark-up$\begin{array}{rll} \mathrm{amount of mark-up} & = & \mathrm{mark-up rate}\cdot\mathrm{original cost} \\ \mathrm{list price} & = & \mathrm{original cost}\,+\,\mathrm{mark up} \\ \end{array}$

• Simple Interest If an amount of money, P, called the principal, is invested or borrowed for a period of t years at an annual interest rate r, the amount of interest, I, earned or paid is:

• How To Solve Geometry Applications Step 1. Read the problem and make sure all the words and ideas are understood. Step 2. Identify what you are looking for. Step 3. Name what you are looking for by choosing a variable to represent it. Draw the figure and label it with the given information. Step 4. Translate into an equation by writing the appropriate formula or model 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.

• The Pythagorean Theorem In any right triangle, where a and b are the lengths of the legs, and c is the length of the hypotenuse, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse.

• Total Value of CoinsFor the same type of coin, the total value of a number of coins is found by using the model$number\cdot{}value=total\,value$ number is the number of coins value is the value of each coin total value is the total value of all the coins

• How to solve coin word problems. Step 1. Read the problem. Make sure all the words and ideas are understood.Determine the types of coins involved.Create a table to organize the information. Label the columns “type,” “number,” “value,” “total value.” List the types of coins. Write in the value of each type of coin. Write in the total value of all the coins. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity.Use variable expressions to represent the number of each type of coin and write them in the table.Multiply the number times the value to get the total value of each type of coin. Step 4. Translate into an equation.It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation.Write the equation by adding the total values of all the types of coins. 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.

• How To Solve a Uniform Motion Application Step 1. Read the problem. Make sure all the words and ideas are understood.Draw a diagram to illustrate what it happening.Create a table to organize the information. Label the columns rate, time, distance. List the two scenarios. Write in the information you know. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity.Complete the chart.Use variable expressions to represent that quantity in each row.Multiply the rate times the time to get the distance. Step 4. Translate into an equation.Restate the problem in one sentence with all the important information.Then, translate the sentence into an equation. 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.

• Inequalities, Number Lines, and Interval Notation$x>a\,x\ge{}a\,x<a\,x\le{}a$

• Linear Inequality A linear inequality is an inequality in one variable that can be written in one of the following forms where a, b, and c are real numbers and $a\ne{}0:$

• Addition and Subtraction Property of Inequality For any numbers a, b, and c, if $a<b,\,\mathrm{then}$

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.

• Multiplication and Division Property of Inequality For any numbers a, b, and c,$\begin{array}{l} \mathrm{multiply or divide by a}\,\mathrm{positive} \\ \,\mathrm{if}\,a<b\,\mathrm{and}\,c>0,\,\mathrm{then}\,ac<bc\,\mathrm{and}\,\frac{a}{c}<\frac{b}{c}. \\ \,\mathrm{if}\,a>b\,\mathrm{and}\,c>0,\,\mathrm{then}\,ac>bc\,\mathrm{and}\,\frac{a}{c}>\frac{b}{c}. \\ \mathrm{multiply or divide by a}\,\mathrm{negative} \\ \,\mathrm{if}\,a<b\,\mathrm{and}\,c<0,\,\mathrm{then}\,ac>bc\,\mathrm{and}\,\frac{a}{c}>\frac{b}{c}. \\ \,\mathrm{if}\,a>b\,\mathrm{and}\,c<0,\,\mathrm{then}\,ac<bc\,\mathrm{and}\,\frac{a}{c}<\frac{b}{c}. \\ \end{array}$

• Phrases that indicate inequalities $>$$\ge$$<$$\le$is greater thanis more thanis larger thanexceeds is greater than or equal tois at leastis no less thanis the minimum is less thanis smaller thanhas fewer thanis lower than is less than or equal tois at mostis no more thanis the maximum

• How to solve a compound inequality with “and” Step 1. Solve each inequality. Step 2. Graph each solution. Then graph the numbers that make both inequalities true. This graph shows the solution to the compound inequality. Step 3. Write the solution in interval notation.

• Double Inequality A double inequality is a compound inequality such as $a<x<b$. It is equivalent to $a<x$ and $x<b.$$\begin{array}{lcclcccccccccccccccccccccccccccccccccccccccccccccccc} \mathrm{Other forms:} & & & \begin{array}{lcclcclcclccl} a<x<b & & & \mathrm{is equivalent to} & & & a<x & & & \mathrm{and} & & & x<b \\ a\le{}x\le{}b & & & \mathrm{is equivalent to} & & & a\le{}x & & & \mathrm{and} & & & x\le{}b \\ a>x>b & & & \mathrm{is equivalent to} & & & a>x & & & \mathrm{and} & & & x>b \\ a\ge{}x\ge{}b & & & \mathrm{is equivalent to} & & & a\ge{}x & & & \mathrm{and} & & & x\ge{}b \\ \end{array} \\ \end{array}$

• How to solve a compound inequality with “or” Step 1. Solve each inequality. Step 2. Graph each solution. Then graph the numbers that make either inequality true. Step 3. Write the solution in interval notation.

• Absolute Value The absolute value of a number is its distance from 0 on the number line. The absolute value of a number n is written as $|n|$ and $|n|\ge{}0$ for all numbers. Absolute values are always greater than or equal to zero.

• Absolute Value Equations For any algebraic expression, u, and any positive real number, a,$\begin{array}{lcccl} \mathrm{if} & & & & |u|=a \\ \mathrm{then} & & & & \,u=-a\,\mathrm{or}\,u=a \\ \end{array}$ Remember that an absolute value cannot be a negative number.

• How to Solve Absolute Value Equations Step 1. Isolate the absolute value expression. Step 2. Write the equivalent equations. Step 3. Solve each equation. Step 4. Check each solution.

• Equations with Two Absolute Values For any algebraic expressions, u and v,$\begin{array}{lcccl} \mathrm{if} & & & & |u|=|v| \\ \mathrm{then} & & & & \,u=-v\,\mathrm{or}\,u=v \\ \end{array}$

• Absolute Value Inequalities with $<$ or $\le$ For any algebraic expression, u, and any positive real number, a,$\begin{array}{lcccclccccl} \mathrm{if} & & & & & |u|<a, & & & & & \mathrm{then}\,-a<u<a \\ \mathrm{if} & & & & & |u|\le{}a, & & & & & \mathrm{then}\,-a\le{}u\le{}a \\ \end{array}$

• How To Solve Absolute Value Inequalities with $<$ or $\le$ Step 1. Isolate the absolute value expression. Step 2. Write the equivalent compound inequality.$\begin{array}{ccccccccccc} |u|<a & & & & & \mathrm{is equivalent to} & & & & & -a<u<a \\ |u|\le{}a & & & & & \mathrm{is equivalent to} & & & & & -a\le{}u\le{}a \\ \end{array}$ Step 3. Solve the compound inequality. Step 4. Graph the solution Step 5. Write the solution using interval notation

• Absolute Value Inequalities with $>$ or $\ge$ For any algebraic expression, u, and any positive real number, a,$\begin{array}{lcccclccccl} \mathrm{if} & & & & & |u|>a, & & & & & \mathrm{then}\,u<-a\,\mathrm{or}\,u>a \\ \mathrm{if} & & & & & |u|\ge{}a, & & & & & \mathrm{then}\,u\le-a\,\mathrm{or}\,u\ge{}a \\ \end{array}$

• How To Solve Absolute Value Inequalities with $>$ or $\ge$ Step 1. Isolate the absolute value expression. Step 2. Write the equivalent compound inequality.$\begin{array}{ccccccccc} |u|>a & & & & \mathrm{is equivalent to} & & & & u<-a\,\mathrm{or}\,u>a \\ |u|\ge{}a & & & & \mathrm{is equivalent to} & & & & u\le-a\,\mathrm{or}\,u\ge{}a \\ \end{array}$ Step 3. Solve the compound inequality. Step 4. Graph the solution Step 5. Write the solution using interval notation

##

Review Exercises

Solve Equations Using the General Strategy for Solving Linear Equations

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

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 or Decimal Coefficients

In the following exercises, solve each equation.

Use a Problem Solving Strategy for Word Problems

In the following exercises, solve using the problem solving strategy for word problems.

Solve Number Word Problems

In the following exercises, solve each number word problem.

Solve Percent Applications

In the following exercises, translate and solve.

In the following exercises, solve.

Solve Simple Interest Applications

In the following exercises, solve.

Solve a Formula for a Specific Variable

In the following exercises, solve the formula for the specified variable.

Use Formulas to Solve Geometry Applications

In the following exercises, solve using a geometry formula.

Solve Coin Word Problems

In the following exercises, solve.

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

Solve Mixture Word Problems

In the following exercises, solve.

Solve Uniform Motion Applications

In the following exercises, solve.

Graph Inequalities on the Number Line

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

Solve Linear Inequalities

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

Translate Words 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.

Solve Applications with Linear Inequalities

In the following exercises, solve.

Solve Compound Inequalities with “and”

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

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.

Solve Applications with Compound Inequalities

In the following exercises, solve.

Solve Absolute Value Equations

In the following exercises, solve.

Solve Absolute Value Inequalities with “less than”

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

Solve Absolute Value Inequalities with “greater than”

In the following exercises, solve. Graph the solution and write the solution in interval notation.

Solve Applications with Absolute Value

In the following exercises, solve.

##

In the following exercises, solve each equation.

In the following exercises, graph the inequality 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

compound inequality

A compound inequality is made up of two inequalities connected by the word “and” or the word “or.”

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.

linear equation

a

b

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

• How 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 equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

• How to Solve Linear Equations Using a General Strategy 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 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 to make sure the result is a true statement.

• How to Solve Equations with Fraction or Decimal Coefficients Step 1. Find the least common denominator (LCD) of all the fractions and decimals (in fraction form) in the equation. Step 2. Multiply both sides of the equation by that LCD. This clears the fractions and decimals. Step 3. Solve using the General Strategy for Solving Linear Equations.

• How To Use a Problem Solving Strategy for Word Problems Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what you are looking for. Step 3. Name what you 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 all the important information. Then, translate the English sentence into an algebra equation. Step 5. Solve the equation using proper algebra techniques. Step 6. Check the answer in the problem to make sure it makes sense. Step 7. Answer the question with a complete sentence.

• How To Find Percent Change Step 1. Find the amount of change$\mathrm{change}=\mathrm{new amount}-\mathrm{original amount}$ Step 2. Find what percent the amount of change is of the original amount.$\mathrm{change is what percent of the original amount?}$

• Discount$\begin{array}{rll} \mathrm{amount of discount} & = & \mathrm{discount rate}\cdot\mathrm{original price} \\ \mathrm{sale price} & = & \mathrm{original amount}-\mathrm{discount} \\ \end{array}$

• Mark-up$\begin{array}{rll} \mathrm{amount of mark-up} & = & \mathrm{mark-up rate}\cdot\mathrm{original cost} \\ \mathrm{list price} & = & \mathrm{original cost}\,+\,\mathrm{mark up} \\ \end{array}$

• Simple Interest If an amount of money, P, called the principal, is invested or borrowed for a period of t years at an annual interest rate r, the amount of interest, I, earned or paid is:

• How To Solve Geometry Applications Step 1. Read the problem and make sure all the words and ideas are understood. Step 2. Identify what you are looking for. Step 3. Name what you are looking for by choosing a variable to represent it. Draw the figure and label it with the given information. Step 4. Translate into an equation by writing the appropriate formula or model 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.

• The Pythagorean Theorem In any right triangle, where a and b are the lengths of the legs, and c is the length of the hypotenuse, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse.

• Total Value of CoinsFor the same type of coin, the total value of a number of coins is found by using the model$number\cdot{}value=total\,value$ number is the number of coins value is the value of each coin total value is the total value of all the coins

• How to solve coin word problems. Step 1. Read the problem. Make sure all the words and ideas are understood.Determine the types of coins involved.Create a table to organize the information. Label the columns “type,” “number,” “value,” “total value.” List the types of coins. Write in the value of each type of coin. Write in the total value of all the coins. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity.Use variable expressions to represent the number of each type of coin and write them in the table.Multiply the number times the value to get the total value of each type of coin. Step 4. Translate into an equation.It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation.Write the equation by adding the total values of all the types of coins. 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.

• How To Solve a Uniform Motion Application Step 1. Read the problem. Make sure all the words and ideas are understood.Draw a diagram to illustrate what it happening.Create a table to organize the information. Label the columns rate, time, distance. List the two scenarios. Write in the information you know. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity.Complete the chart.Use variable expressions to represent that quantity in each row.Multiply the rate times the time to get the distance. Step 4. Translate into an equation.Restate the problem in one sentence with all the important information.Then, translate the sentence into an equation. 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.

• Inequalities, Number Lines, and Interval Notation$x>a\,x\ge{}a\,x<a\,x\le{}a$

• Linear Inequality A linear inequality is an inequality in one variable that can be written in one of the following forms where a, b, and c are real numbers and $a\ne{}0:$

• Addition and Subtraction Property of Inequality For any numbers a, b, and c, if $a<b,\,\mathrm{then}$

We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.

• Multiplication and Division Property of Inequality For any numbers a, b, and c,$\begin{array}{l} \mathrm{multiply or divide by a}\,\mathrm{positive} \\ \,\mathrm{if}\,a<b\,\mathrm{and}\,c>0,\,\mathrm{then}\,ac<bc\,\mathrm{and}\,\frac{a}{c}<\frac{b}{c}. \\ \,\mathrm{if}\,a>b\,\mathrm{and}\,c>0,\,\mathrm{then}\,ac>bc\,\mathrm{and}\,\frac{a}{c}>\frac{b}{c}. \\ \mathrm{multiply or divide by a}\,\mathrm{negative} \\ \,\mathrm{if}\,a<b\,\mathrm{and}\,c<0,\,\mathrm{then}\,ac>bc\,\mathrm{and}\,\frac{a}{c}>\frac{b}{c}. \\ \,\mathrm{if}\,a>b\,\mathrm{and}\,c<0,\,\mathrm{then}\,ac<bc\,\mathrm{and}\,\frac{a}{c}<\frac{b}{c}. \\ \end{array}$

• Phrases that indicate inequalities $>$$\ge$$<$$\le$is greater thanis more thanis larger thanexceeds is greater than or equal tois at leastis no less thanis the minimum is less thanis smaller thanhas fewer thanis lower than is less than or equal tois at mostis no more thanis the maximum

• How to solve a compound inequality with “and” Step 1. Solve each inequality. Step 2. Graph each solution. Then graph the numbers that make both inequalities true. This graph shows the solution to the compound inequality. Step 3. Write the solution in interval notation.

• Double Inequality A double inequality is a compound inequality such as $a<x<b$. It is equivalent to $a<x$ and $x<b.$$\begin{array}{lcclcccccccccccccccccccccccccccccccccccccccccccccccc} \mathrm{Other forms:} & & & \begin{array}{lcclcclcclccl} a<x<b & & & \mathrm{is equivalent to} & & & a<x & & & \mathrm{and} & & & x<b \\ a\le{}x\le{}b & & & \mathrm{is equivalent to} & & & a\le{}x & & & \mathrm{and} & & & x\le{}b \\ a>x>b & & & \mathrm{is equivalent to} & & & a>x & & & \mathrm{and} & & & x>b \\ a\ge{}x\ge{}b & & & \mathrm{is equivalent to} & & & a\ge{}x & & & \mathrm{and} & & & x\ge{}b \\ \end{array} \\ \end{array}$

• How to solve a compound inequality with “or” Step 1. Solve each inequality. Step 2. Graph each solution. Then graph the numbers that make either inequality true. Step 3. Write the solution in interval notation.

• Absolute Value The absolute value of a number is its distance from 0 on the number line. The absolute value of a number n is written as $|n|$ and $|n|\ge{}0$ for all numbers. Absolute values are always greater than or equal to zero.

• Absolute Value Equations For any algebraic expression, u, and any positive real number, a,$\begin{array}{lcccl} \mathrm{if} & & & & |u|=a \\ \mathrm{then} & & & & \,u=-a\,\mathrm{or}\,u=a \\ \end{array}$ Remember that an absolute value cannot be a negative number.

• How to Solve Absolute Value Equations Step 1. Isolate the absolute value expression. Step 2. Write the equivalent equations. Step 3. Solve each equation. Step 4. Check each solution.

• Equations with Two Absolute Values For any algebraic expressions, u and v,$\begin{array}{lcccl} \mathrm{if} & & & & |u|=|v| \\ \mathrm{then} & & & & \,u=-v\,\mathrm{or}\,u=v \\ \end{array}$

• Absolute Value Inequalities with $<$ or $\le$ For any algebraic expression, u, and any positive real number, a,$\begin{array}{lcccclccccl} \mathrm{if} & & & & & |u|<a, & & & & & \mathrm{then}\,-a<u<a \\ \mathrm{if} & & & & & |u|\le{}a, & & & & & \mathrm{then}\,-a\le{}u\le{}a \\ \end{array}$

• How To Solve Absolute Value Inequalities with $<$ or $\le$ Step 1. Isolate the absolute value expression. Step 2. Write the equivalent compound inequality.$\begin{array}{ccccccccccc} |u|<a & & & & & \mathrm{is equivalent to} & & & & & -a<u<a \\ |u|\le{}a & & & & & \mathrm{is equivalent to} & & & & & -a\le{}u\le{}a \\ \end{array}$ Step 3. Solve the compound inequality. Step 4. Graph the solution Step 5. Write the solution using interval notation

• Absolute Value Inequalities with $>$ or $\ge$ For any algebraic expression, u, and any positive real number, a,$\begin{array}{lcccclccccl} \mathrm{if} & & & & & |u|>a, & & & & & \mathrm{then}\,u<-a\,\mathrm{or}\,u>a \\ \mathrm{if} & & & & & |u|\ge{}a, & & & & & \mathrm{then}\,u\le-a\,\mathrm{or}\,u\ge{}a \\ \end{array}$

• How To Solve Absolute Value Inequalities with $>$ or $\ge$ Step 1. Isolate the absolute value expression. Step 2. Write the equivalent compound inequality.$\begin{array}{ccccccccc} |u|>a & & & & \mathrm{is equivalent to} & & & & u<-a\,\mathrm{or}\,u>a \\ |u|\ge{}a & & & & \mathrm{is equivalent to} & & & & u\le-a\,\mathrm{or}\,u\ge{}a \\ \end{array}$ Step 3. Solve the compound inequality. Step 4. Graph the solution Step 5. Write the solution using interval notation

##

Review Exercises

Solve Equations Using the General Strategy for Solving Linear Equations

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

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 or Decimal Coefficients

In the following exercises, solve each equation.

Use a Problem Solving Strategy for Word Problems

In the following exercises, solve using the problem solving strategy for word problems.

Solve Number Word Problems

In the following exercises, solve each number word problem.

Solve Percent Applications

In the following exercises, translate and solve.

In the following exercises, solve.

Solve Simple Interest Applications

In the following exercises, solve.

Solve a Formula for a Specific Variable

In the following exercises, solve the formula for the specified variable.

Use Formulas to Solve Geometry Applications

In the following exercises, solve using a geometry formula.

Solve Coin Word Problems

In the following exercises, solve.

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

Solve Mixture Word Problems

In the following exercises, solve.

Solve Uniform Motion Applications

In the following exercises, solve.

Graph Inequalities on the Number Line

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

Solve Linear Inequalities

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

Translate Words 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.

Solve Applications with Linear Inequalities

In the following exercises, solve.

Solve Compound Inequalities with “and”

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

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.

Solve Applications with Compound Inequalities

In the following exercises, solve.

Solve Absolute Value Equations

In the following exercises, solve.

Solve Absolute Value Inequalities with “less than”

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

Solve Absolute Value Inequalities with “greater than”

In the following exercises, solve. Graph the solution and write the solution in interval notation.

Solve Applications with Absolute Value

In the following exercises, solve.

##

In the following exercises, solve each equation.

In the following exercises, graph the inequality 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. $-12n+5=8n,n=-\frac{5}{4}$

2. $-(s+4)=18$

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. $\frac{2}{5}n-\frac{1}{10}=\frac{7}{10}$

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

5. $0.8x-0.3=0.7x+0.2$

1. Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?

2. One number is two more than four times another. Their sum is negative thirteen. Find the numbers.

3. Find three consecutive even integers whose sum is 234.

4. Koji has $5,502 in his savings account. This is$30 less than six times the amount in his checking account. How much money does Koji have in his checking account?

5. The bill for Dino’s lunch was $19.45. He wanted to leave 20% of the total bill as a tip. How much should the tip be?

1. Jaden earns $2,680 per month. He pays$938 a month for rent. What percent of his monthly pay goes to rent?

2. Rowena’s monthly gasoline bill dropped from $83.75 last month to$56.95 this month. Find the percent change.

3. Lacey bought a pair of boots on sale for $95. The original price of the boots was$200. Find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent, if needed.)

4. Winston deposited $3,294 in a bank account with interest rate 2.6% How much interest was earned in five years?

5. Jaime’s refrigerator loan statement said he would pay $1,026 in interest for a four-year loan at 13.5%. How much did Jaime borrow to buy the refrigerator?

1. Solve the formula$A=\frac{1}{2}{d}_{1}{d}_{2}$ for ${d}_{2}.$

2. Solve the formula$4x-3y=12$ for y.

3. The measure of the smallest angle in a right triangle is $45^{\circ}$ less than the measure of the next larger angle. Find the measures of all three angles.

4. Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is eight feet tall and Sergio has 10 feet of wire. How far from the base of the antenna can he attach the wire? Approximate to the nearest tenth, if necessary.

5. The length of a rectangle is 12 cm more than the width. The perimeter is 74 cm. Find the length and the width.

1. The perimeter of a triangle is 35 feet. One side of the triangle is five feet longer than the second side. The third side is three feet longer than the second side. Find the length of each side.

2. Lenny has $3.69 in pennies, dimes, and quarters. The number of pennies is three more than the number of dimes. The number of quarters is twice the number of dimes. How many of each coin does he have?

3. 125 tickets were sold for the jazz band concert for a total of $1,022. Student tickets cost$6 each and general admission tickets cost $10 each. How many of each kind of ticket were sold?

4. Marquese is making 10 pounds of trail mix from raisins and nuts. Raisins cost $3.45 per pound and nuts cost$7.95 per pound. How many pounds of raisins and how many pounds of nuts should Marquese use for the trail mix to cost him $6.96 per pound?

5. Enrique borrowed $23,500 to buy a car. He pays his uncle 2$4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total $23,500? (Round your answer to the nearest tenth of a percent.)

1. Louellen and Tracy met at a restaurant on the road between Chicago and Nashville. Louellen had left Chicago and drove 3.2 hours towards Nashville. Tracy had left Nashville and drove 4 hours towards Chicago, at a speed one mile per hour faster than Louellen’s speed. The distance between Chicago and Nashville is 472 miles. Find Louellen’s speed and Tracy’s speed.

2. Kyle rowed his boat upstream for 50 minutes. It took him 30 minutes to row back downstream. His speed going upstream is two miles per hour slower than his speed going downstream. Find Kyle’s upstream and downstream speeds.

3. Anthony drove from New York City to Baltimore, which is a distance of 192 miles. He left at 3:45 and had heavy traffic until 5:30. Traffic was light for the rest of the drive, and he arrived at 7:30. His speed in light traffic was four miles per hour more than twice his speed in heavy traffic. Find Anthony’s driving speed in heavy traffic and light traffic.

4. $x\ge-2.5$

5. $x>2$

1. $-5\le{}x<-3$

2. $n-12\le{}23$

3. $9x>54$

4. $6p>15p-30$

5. $5n-15(4-n)\le{}10(n-6)+10n$

1. Five more than z is at most 19.

2. Nine times n exceeds 42.

3. Julianne has a weekly food budget of $231 for her family. If she plans to budget the same amount for each of the seven days of the week, what is the maximum amount she can spend on food each day?

4. Briana has been offered a sales job in another city. The offer was for $42,500 plus 8$66,500. What would her total sales need to be for her to move?

5. Costa is an accountant. During tax season, he charges $125 to do a simple tax return. His expenses for buying software, renting an office, and advertising are$6,000. How many tax returns must he do if he wants to make a profit of at least $8,000?

1. $x\le{}5$ and $x>-3$

2. $5(3x-2)\le{}5$ and$4(x+2)<3$

3. $\frac{3}{4}x-5\ge-2$ and$-3(x+1)\ge{}6$

4. $5-2x\le-1$ or$6+3x\le{}4$

5. $\frac{3}{4}x-2>4$ or $4(2-x)>0$

1. $\frac{1}{2}x-3\le{}4$ or$\frac{1}{3}(x-6)\ge-2$

2. 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.

3. $|y|=-14$

4. $|3x-4|+5=7$

5. $-2|x-3|+8=-4$

1. $|6x-5|=|2x+3|$

2. $|2x-5|\le{}3$

3. $|5x+1|\le-2$

4. $|x|\ge{}2$

5. $|x-7|\ge{}1$

1. A craft beer brewer needs 215,000 bottle per day. But this total can vary by as much as 5,000 bottles. What is the maximum and minimum expected usage at the bottling company?

2. $-5(2x+1)=45$

3. $8(3a+5)-7(4a-3)=20-3a$

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

5. $\frac{3}{4}x-\frac{2}{3}=\frac{1}{2}x+\frac{5}{6}$

1. $|2x-1|=|4x+3|$

2. $x\ge-3.5$

3. $-2\le{}x<5$

4. $3c-10(c-2)<5c+16$

5. $3(2x-3)<-5$ or$4x-1>3$

1. $|4x-3|\ge{}5$

2. Find the length of the missing side.

3. The sum of two consecutive odd integers is $-112.$ Find the numbers.

4. Bonita has $2.95 in dimes and quarters in her pocket. If she has five more dimes than quarters, how many of each coin does she have?

5. The measure of one angle of a triangle is twice the measure of the smallest angle. The measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.

1. Two planes leave Dallas at the same time. One heads east at a speed of 428 miles per hour. The other plane heads west at a speed of 382 miles per hour. How many hours will it take them to be 2,025 miles apart?

2. Sara has a budget of $1,000 for costumes for the 18 members of her musical theater group. If all the costumes are the same price, what is the maximum she can spend for each costume?