## Absolute Value Equations and Inequalities

*August 24, 2011 at 10:58 am* *
2 comments *

I have just finished up a unit in my Intermediate Algebra class on absolute value equations & inequalities. This post is the first in a series of 3 on solving these equations and inequalities. I’ll start with strategies today to rewrite the equation or inequality without absolute values.

**Behind the Scenes**

Consider the following.

Three similar statements comparing the absolute value of *x* to the positive number 5.

**Equations**

On a number line, the solutions to are the two points and . (Graph from Wolfram|Alpha.)

So, the equation is equivalent to or .

If you have an equation of the form , where X is any algebraic linear expression and *a* is a positive number, begin by converting the equation to the two linear equations: or .

Example: or

**Inequalities, Less Than**

If you look at all the values between -5 & 5, you’ll notice that the one thing they have in common is that their absolute values are all less than 5. Recall that is the distance between *x* and 0 on the number line. Since all of the points between -5 and 5 are less than 5 units away from 0 on the number line, these are the solutions to .

So, is equivalent to .

If you have an inequality of the form , where X is any algebraic linear expression and *a* is a positive number, begin by converting the inequality to the compound inequality: .

Example:

**Inequalities, Greater Than**

Finally, all of the values outside of this region are more than 5 units away from 0, meaning that they are all solutions to .

So, is equivalent to or .

If you have an inequality of the form , where X is any algebraic linear expression and *a* is a positive number, begin by converting the inequality to the compound inequality: or .

Example: or

**Overall Strategy**

- Begin by isolating the absolute value on the left side of the equation or inequality.
- To “get rid of” the absolute values, use the ideas from above. The step to do this really depends on the sign: <, >, or =.
- Solve the new equations or inequalities.

In the next blog I will go over strategies when the number *a* is a negative number.

– George

*I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.*

Entry filed under: General Teaching, Math. Tags: absolute value, absolute value equations, absolute value inequalities, algebra, developmental math, george woodbury, intermediate algebra, Math, teaching, Wolfram Alpha, woodbury.

1.David Radcliffe (@daveinstpaul) | August 24, 2011 at 11:24 amPerhaps this is a nitpick, but I would use a double arrow instead of the single arrow –> to connect equivalent statements. |2x – 1| = 7 if and only if 2x – 1 = 7 or 2x – 1 = -7, so the double arrow is most appropriate here.

2.georgewoodbury | August 24, 2011 at 12:01 pmNot a nitpick, meant to include the left arrow too! Editing on the way. Thanks David.