# Solving Linear Equations

## Solve Absolute Value Inequalities


### Solve Absolute Value Equations

As we prepare to solve absolute value equations, we review our definition of absolute value.
We learned that both a number and its opposite are the same distance from zero on the number line. Since they have the same distance from zero, they have the same absolute value. For example:
 is 5 units away from 0, so 
 is 5 units away from 0, so 
 illustrates this idea.
For the equation  we are looking for all numbers that make this a true statement. We are looking for the numbers whose distance from zero is 5. We just saw that both 5 and  are five units from zero on the number line. They are the solutions to the equation.
The solution can be simplified to a single statement by writing  This is read, “x is equal to positive or negative 5”.
We can generalize this to the following property for absolute value equations.
To solve an absolute value equation, we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.
The steps for solving an absolute value equation are summarized here.
Remember, an absolute value is always positive!
Some of our absolute value equations could be of the form  where u and v are algebraic expressions. For example, 
How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, u, and a positive real number, a, if  then  or 
This tells us that
This leads us to the following property for equations with two absolute values.
When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.

### Solve Absolute Value Inequalities with “Less Than”

Let’s look now at what happens when we have an absolute value inequality. Everything we’ve learned about solving inequalities still holds, but we must consider how the absolute value impacts our work.
Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line. For the equation  we saw that both 5 and  are five units from zero on the number line. They are the solutions to the equation.
What about the inequality  Where are the numbers whose distance is less than or equal to 5? We know  and 5 are both five units from zero. All the numbers between  and 5 are less than five units from zero. See .
In a more general way, we can see that if  then  See .
This result is summarized here.
After solving an inequality, it is often helpful to check some points to see if the solution makes sense. The graph of the solution divides the number line into three sections. Choose a value in each section and substitute it in the original inequality to see if it makes the inequality true or not. While this is not a complete check, it often helps verify the solution.

### Solve Absolute Value Inequalities with “Greater Than”

What happens for absolute value inequalities that have “greater than”? Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line.
We started with the inequality  We saw that the numbers whose distance is less than or equal to five from zero on the number line were  and 5 and all the numbers between  and 5. See .
Now we want to look at the inequality  Where are the numbers whose distance from zero is greater than or equal to five?
Again both  and 5 are five units from zero and so are included in the solution. Numbers whose distance from zero is greater than five units would be less than  and greater than 5 on the number line. See .
In a more general way, we can see that if  then  or  See .
This result is summarized here.

### Solve Applications with Absolute Value

Absolute value inequalities are often used in the manufacturing process. An item must be made with near perfect specifications. Usually there is a certain tolerance of the difference from the specifications that is allowed. If the difference from the specifications exceeds the tolerance, the item is rejected.

### Key Concepts

1. 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  and  for all numbers.
Absolute values are always greater than or equal to zero.
2. Absolute Value Equations
For any algebraic expression, u, and any positive real number, a,
Remember that an absolute value cannot be a negative number.
3. How to Solve Absolute Value Equations


4. Equations with Two Absolute Values
For any algebraic expressions, u and v,
5. Absolute Value Inequalities with  or 
For any algebraic expression, u, and any positive real number, a,
6. How To Solve Absolute Value Inequalities with  or 


7. Absolute Value Inequalities with  or 
For any algebraic expression, u, and any positive real number, a,
8. How To Solve Absolute Value Inequalities with  or 



### Section Exercises


### Practice Makes Perfect

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 each inequality. Graph the solution and write the solution in interval notation.
In the following exercises, solve. For each inequality, also graph the solution and write the solution in interval notation.
Solve Applications with Absolute Value
In the following exercises, solve.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

### Chapter Review Exercises


### Use a General Strategy to Solve Linear Equations

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

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

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 Mixture and Uniform Motion Applications

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.

### Solve Linear Inequalities

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

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 Inequalities

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.

### Practice Test

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.