# 2.7 Solve Absolute Value Inequalities

Topics covered in this section are:

## 2.7.1 Solve Absolute Value Equations

As we prepare to solve absolute value equations, we review our definition of absolute value.

### ABSOLUTE VALUE

The absolute value of a number is its distance from zero on the number line.

The absolute value of a number $n$ is written as $|n|$ and $|n|≥0$ for all numbers.

Absolute values are always greater than or equal to zero.

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:

$-5$ is $5$ units away from $0$, so $|-5| = 5$.

$5$ is $5$ units away from $0$, so $|5| = 5$.

Figure 2.6 illustrates this idea. Figure 2.6 The numbers $5$ and $-5$ are both five units away from zero.

For the equation $|x|=5$, 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 $-5$ are five units from zero on the number line. They are the solutions to the equation.

 If $|x|=5$ then $x=-5$ or $x=5$

The solution can be simplified to a single statement by writing $x±5$. This is read, “$x$ is equal to positive or negative $5$”.

We can generalize this to the following property for absolute value equations.

### ABSOLUTE VALUE EQUATIONS

For any algebraic expression, $u$, and any positive real number, $a$,

 if $|u|=a$ then $u=-a$ or $u=a$

Remember that an absolute value cannot be a negative number.

#### Example 1

Solve:

• $|x|=8$
• $|y|=-6$
• $|z|=0$
Solution

Part 1.

Part 2.

Part 3.

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.

#### Example 2

Solve $|5x-4|-3=8$.

Solution

The steps for solving an absolute value equation are summarized here.

### HOW TO: Solve absolute value equations.

1. Isolate the absolute value expression.
2. Write the equivalent equations.
3. Solve each equation.
4. Check each solution.

#### Example 3

Solve $2|x-7|+5=9$.

Solution

Remember, an absolute value is always positive!

#### Example 4

Solve: $| \frac{2}{3}x-4|+11=3$.

Solution

Some of our absolute value equations could be of the form $|u|=|v|$ where $u$ and $v$ are algebraic expressions. For example, $|x-3|=|2x+1|$.

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 $|u|=a$, then $u=-a$ or $u=a$.

This tells us that:

 if $|u|=|v|$ then $u=-v$ or $u=v$

This leads us to the following property for equations with two absolute values.

### EQUATIONS WITH TWO ABSOLUTE VALUES

For any algebraic expressions, $u$ and $v$,

 if $|u|=|v|$ then $u=-v$ or $u=v$

When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.

#### Example 5

Solve: $|5x-1|=|2x+3|$.

Solution

## 2.7.2 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 $|x|=5$, we saw that both $5$ and $-5$ are five units from zero on the number line. They are the solutions to the equation.

$|x|=5$

$x=-5 \ \$ or $\ \ x=5$

What about the inequality $|x|≤5$? Where are the numbers whose distance is less than or equal to $5$? We know $-5$ and $5$ are both five units from zero. All the numbers between $-5$ and $5$ are less than five units from zero. See Figure 2.7.

In a more general way, we can see that if $|u|≤a$, then $-a≤u≤a$. See Figure 2.8.

This result is summarized here.

### ABSOLUTE VALUE INEQUALITIES WITH $<$ OR $≤$

For any algebraic expression, $u$, and any positive real number, $a$,

Solution