## Absolute Value Equations and Inequalities

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.

• $|x|=5$
• $|x|<5$
• $|x|>5$

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

Equations

On a number line, the solutions to $|x|=5$ are the two points $x=5$ and $x=-5$. (Graph from Wolfram|Alpha.)

So, the equation $|x|=5$ is equivalent to $x=5$ or $x=-5$.

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

Example: $|2x-1|=7$ $\Leftrightarrow$ $2x-1=7$ or $2x-1=-7$

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 $|x|$ 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 $|x|<5$.

So, $|x|<5$ is equivalent to $-5.

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

Example: $|3x+4|<23$ $\Leftrightarrow$ $-23<3x+4<23$

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 $|x|>5$.

So, $|x>5$ is equivalent to $x>5$ or $x<-5$.

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

Example: $|x+7|>15$ $\Leftrightarrow$ $x+7>15$ or $x+7<-15$

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.

• 1. David Radcliffe (@daveinstpaul)  |  August 24, 2011 at 11:24 am

Perhaps 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 pm

Not a nitpick, meant to include the left arrow too! Editing on the way. Thanks David.