Absolute Value Equations and Inequalities

Absolute Value Equations

Absolute value is the distance between a number and zero on a number line. Since absolute value represents a distance, its result is always positive. The symbol for absolute value is a pair of bars around the variable or number: |a|. Absolute value is often used in situations where answers involve a range of numbers.

First, look at this very simple absolute value equation:

On a number line, it would like this: 

The answer to this equation is x = 3 or x = -3. These are the two values that would make the equation true, since taking the absolute value of a negative number gives a positive result.

There is no solution to the problem since there are no numbers that you can put in place of x to make an answer of -4.

To solve absolute value equations, be sure to get the absolute value alone on one side of the equation.

Example

m = 5 or -5. This answer can also be written as .

n = 9 or -9

Check your work by putting the values of the variable back into the original equation.

What happens when the variable isn’t alone inside the absolute-value bars? In those cases, you will solve two quick equations instead of one because you have to set the absolute value equal to the positive and negative numbers. You will end up with two answers.

Example

p + 2 = 9 or p + 2 = -9

p = 7 or p = -11.

This equation works because we are looking for the numbers that will create a sum whose absolute value is nine. Check the answers to be sure they are correct:

If there are other numbers outside the absolute value on the same side of the equation, work to get the absolute value part alone before breaking the equation into two parts.

Example

Absolute-Value Inequalities

It would be nice if absolute-value inequalities were as easy to work with as absolute-value equations. Alas, absolute-value inequalities are more complicated.

Absolute-value inequalities become compound inequalities. Compound inequalities are two inequalities joined by AND or OR. In logic and math problems, AND means both or all parts are true. OR means that one or the other part is true, but not necessarily both.

Example

Think of a number that is even AND greater than ten. 12 works, 8 does not. The number must meet both criteria.

Think of a number that is even OR is greater than ten. 8 works, 12 works, and so does 15. The number only needs to meet one of the two criteria, although it can meet both.

Absolute-value inequalities are best explained with the help of examples. Here are the most basic types of inequality problems.

“Less Than” Absolute-Value Inequalities

In “less than” and “less than or equal to” absolute-value inequalities, set the absolute value part of the inequality between the positive and negative values of the rest of the inequality. This is better shown in an example:

-12 < n + 3 < 12

Or as two separate inequalities:

-12 < n + 3 AND n + 3 < 12

Solve by getting the variable alone, the same way that you would in an equation.

n > -15 AND n < 9

Written as a combined inequality, the solution would be:

-15 < n < 9

The only numbers that will make this inequality be true are numbers greater than -15 AND less than 9.

Here is one more example to view:

-25 < k -5 < 25

-20 < k < 30

Values of k between -20 and 30 make the inequality true.

“Greater Than” Absolute-Value Inequalities

In “greater than” and “greater than or equal to” absolute-value inequalities, you have to create inequalities that go in opposite directions. The first inequality will have the absolute value be less than the negative value, and the second inequality will have the absolute value greater than the positive value.

Let’s look at two examples that show how this works.

Write this as two separate inequalities and solve:

OR

OR

Review

• Absolute value is the distance a number is from zero on a number line.
• In absolute-value equations, the variable is equal to either the positive or negative value that the absolute value was equal to. |x| = 7 means that x = 7 or -7.
• When solving absolute-value equations, be sure to isolate the absolute value on one side of the equation before creating the two new equations to solve.
• Compound inequalities are two inequalities joined by and or or.
• For absolute-value inequalities that use the “less than” or “less than or equal” sign: set the absolute value part of the inequality between the positive and negative values of the number from the other side of the inequality. (This is an and compound inequality.)
• For absolute-value inequalities that use the “greater than” or “greater than or equal” sign, make two inequalities. In the first one, set the absolute value part of the inequality to be less than the negative value, and in the second inequality, set the absolute value greater than the positive value. (This is an or compound inequality.)