Solving Absolute Value Inequalities

Lesson Plan - Get It!

Audio:

The opening problem to this lesson seems a little daunting, doesn't it?

Have no worries. Today you are going to learn how to solve absolute value inequalities so you can solve problems just as challenging as that one!

• What is absolute value?
• What is an inequality?

The absolute value of a number is its distance from 0 regardless of which direction (positive or negative). Inequality simply means things are not equal.

(If you need a refresh on these topics, review our lessons found under Additional Resources in the right-hand sidebar first.)

This will be the example problem for this part of the lesson on how to solve an absolute value inequality: |x| (This is a one-step inequality problem).

• First, you need to be aware that the example problem is referred to as an inequality, NOT an equation! An equation will have an equals sign in it (=). An inequality will have a greater than sign (>), a less than sign (

Put the problem into words you can understand. |x| < 12 is the same as saying, "What are all the x's that are less than 12 away from 0?"

Draw a number line like the one below. Put a red dot on the zero:

Now, think for a moment.

• What numbers on this number line have an absolute value of 12?

Exactly! 12 and -12 are both 12 spaces away from 0, so each has an absolute value of 12. To help you finish solving this problem, draw an open circle (not colored in) on 12 and -12.

Look at your number line.

• Not including 12 and -12, what numbers have an absolute value less than 12?
• Another way to think of this is, what numbers are less than 12 spaces away from 0?

Correct! All of the numbers that fall in between -12 and 12 on the number line have an absolute value less than 12 (|x| < 12) To show this on the number line, draw a line between -12 and 12 on your number line.

Now, you have shown the answer to the problem visually, with the help of a number line.

To write it numerically, you would say x > -12 and x < 12.

Finally, the formal way to write the answer to this problem is -12 < x < 12. This is called the solution set. It is the range of numbers on the number line where the answer can be found.

Take a look at your number line where you drew the line between -12 and 12.

Notice that the circles you placed on each of those numbers are open; there is not fill inside like the circle on 0.

The reason for this is that the original problem was asking for < 12. If it were asking for ≤ 12, you could have filled the circles in.

So remember, if the problem is only asking for less than or greater than, your circles will be open (not filled in). If the problem is asking for less than or equal to, or greater than or equal to, then your circles will be closed (filled in).

Take a moment to watch the KHAN Academy video Absolute value inequalities | Linear equations | Algebra | Khan for further instruction on this topic:

If you are getting the hang of this, move on the Got It? section to practice your skills.