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.

  • |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<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: -a<X<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.

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Entry filed under: General Teaching, Math. Tags: , , , , , , , , , , .

Intro Stats – Sampling Techniques Activity Absolute Value Equations & Inequalities: Exceptions (part 2 of 3)

2 Comments Add your own

  • 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.

    Reply
    • 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.

      Reply

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