## Table of Contents

- Absolute Value Inequalities
- Absolute Value Inequality Rules
- Graphing Absolute Value Inequalities
- Lesson Summary

Learn what absolute value inequalities are and how to solve absolute value inequalities. The lesson also covers the graphing of absolute value inequalities.
Updated: 11/11/2021

- Absolute Value Inequalities
- Absolute Value Inequality Rules
- Graphing Absolute Value Inequalities
- Lesson Summary

An **absolute value inequality** is a mathematical expression of two unequal quantities which includes at least one absolute value.

The **absolute value** is the distance of a number from zero. Sometimes this is described as the magnitude. For example the absolute value of 5 is 5 and the absolute value of -5 is also 5. The symbol for absolute value is |n|, where n is a number. Absolute values may also be found for algebraic expressions. The number, variable, or expression within the absolute value symbol is called the argument. In general:

$$|n| = n $$

$$|-n| = n $$

Read: The absolute value of a number n is equal to n and the absolute value of -n is equal to n.

An **inequality** is a mathematical expression stating that one quantity is greater than, greater than or equal to, less than, or less than or equal to another quantity, indicated by the following symbols:

A simple example of an absolute value inequality is $$|x| \geq 7 $$

Following the definition and example above, the absolute value of x is equal to 7 if x is 7 or -7. So The absolute value of x is equal to or greater than 7 if x is any number greater than 7 or less than -7. This is because any number less than -7 will have an absolute value greater than 7.

The solution is written:

$$x\geq 7 $$

or

$$x\leq - 7 $$

This solution is read: X is equal to or greater than 7 OR x is less than or equal to -7

When solving absolute value inequalities there are some important rules to remember:

- Absolute value inequalities give us compound inequalities, a combination of two inequalities.
- Inequalities with a less than symbol have solutions that are intersections of two inequalities, described as "and"
- Inequalities with greater than symbols have solutions that are unions of two inequalities, described as "or"
- When solving inequalities, multiplication or division by a negative number reverses the direction of the inequality symbol
- Since absolute value is positive, an absolute value inequality set less than a negative number will have no solution.
- If the absolute value is greater than any negative number, then the solution is all real numbers

Use these steps to solve absolute value inequalities:

- First use inverse operations to isolate the absolute value on the left side.
- For 'greater than' inequalities, solve for the the inequality greater than the positive and flip the symbol and solve for the the negative value.

For example:

{eq}3|2-x|-5> 7 {/eq}

First,use inverse operations to isolate the absolute value:

Since this is a 'greater than' inequality, solve for the the inequality greater than the positive and flip the symbol and solve for the the negative value:

Solving for the positive option:

Flip the inequality symbol and solve for the negative option:

**The solution is X<-2 or x>6 **

$$|3x+4|\leq25 $$

This is a 'less than' absolute inequality so the solution will be an 'and' complex inequality. The values that make the inequality true must satisfy both conditions: they must be greater than the negative value AND less than the positive value. This type of inequality is written with the argument of the absolute value in the middle of the negative and positive options like this:

$$-25\leq |3x+4|\leq 25 $$

Use inverse operations to solve for x:

**This solution can be read "-29/3 is less than or equal to x and x is less than or equal to 7" or simply "-29/3 is less than or equal to x is less than or equal to 7"**

Here is one more example:

In this case, when the absolute value expression is isolated, it is less than a negative value. Since absolute values are always positive, this inequality has **no solution**

Graphing the solution to an absolute value inequality helps to make sense of the solution.

The solution to one variable inequalities like the examples above can be shown on the the number line.

For the inequality 3|2-x|-5> 7 with the solution x <-2 or x>6, the graph is:

For the inequality above with the "and" solution $$-29/3\leq x\leq 7 $$

the graph is

Inequalities with two variables can be solved efficiently with graphing. Consider the graphs of the one variable inequalities on the number line. Introducing a second variable transforms the graphs into two dimensional regions instead of one dimensional rays or segments. The first step to solving these inequalities is to graph the boundary of the solution region. This is done by graphing the equation of the inequality.

Given the two-variable absolute inequality:

$$y<|x-3|+4 $$

The corresponding equation is $$y=|x-3|+4 $$

The vertex of the graph is at (3, 4) with the slope to the left of the vertex of -1 and slope right of the vertex of -1 giving the following graph:

To show the inequality y<|x-3|+4, the line is show the line as dashed to show that the line itself is not included (because the inequality does not include equal to). The area below the equation graph is shaded because this is the region where y is less than the absolute value expression.

For 'greater than' inequalities, the area above the line is shaded as shown in the following example:

The absolute value is the distance from zero. Since direction doesn't matter, it is always a positive value. Because both the positive or negative value of a quantity is positive, absolute value inequalities have complex solutions made of two inequalities. Absolute value inequalities are solved similarly to equations, using inverse operations with some special rules.

- 'Less than' absolute value inequalities have 'or' solutions while 'greater than' have 'and' solutions.
- When solving 'greater than' inequalities, solve normally for the positive value and flip the inequality symbol for the negative value.
- Flip the inequality symbol when dividing or multiplying by a negative number.
- One variable inequalities are graphed on a number line. 'Or' inequality graphs have two sections in opposite directions, 'and' solutions have one segment.
- Two variable inequality graphs are 2-dimensional areas. Graph the line for the corresponding equation, use a solid line if the the inequality includes 'equal to' and a broken line if it does not, shade in the area above for greater than, shade in the area below for less than.

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Frequently Asked Questions

Solve one dimensional absolute value inequalities algebraically. First, isolate the absolute value on the left using inverse operations. Solve for both the negative and the positive values. For "greater than" inequalities, flip the inequality symbol before solving for the negative value.

Solve two variable absolute value inequalities by graphing them. Graph the corresponding absolute value equation. Use a solid line if the inequality includes 'or equal to' or a broken line if it does not. Shade in the appropriate area above or below the line.

Because each number and its opposite (both positive and negative) have the same absolute value, the absolute value symbol means the inequality will have a two part solution, or a complex inequality. Finding the solution involves solving two inequalities, one for the negative value and one for the positive value.

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