## Absolute Value Equations and Inequalities

These are the best of topics. These are the worst of topics. I love teaching absolute value equations and inequalities because it gives me a chance to focus on conceptual understanding. Every time I gave this lecture I would leave class thinking “That’s it. A math lecture could never be better than that!” Then the exam would come along and my students would struggle with these problems. I just couldn’t understand what was going wrong.

To me, these topics are perfect examples on how important some memorizing can be to gaining understanding. Since I have focused on memorization tips for my students these topics go much better for my students. I start with some conceptual development of the topics. I draw a number line and ask my students to put a dot on the number line at the solutions of the equation |x| = 5. Once we have -5 and 5 on the number line, I ask my students to think about the values between -5 and 5. I ask a student to pick a value, and then ask for the absolute value of that number. Then I ask two more students to do the same with a different value in the interval. I then ask the class to tell me what they all have in common, and someone will tell me that they all have absolute values that are less than 5, and I write |x| < 5 above that interval. I then show them that |x| < 5 is equivalent to -5 < x < 5. I then use the other two intervals to show that |x| > 5 is equivalent to x < -5 or x > 5.

My students have seen how to start to solve absolute value equations and inequalities.

• |x| = a  —->  x = a  or  x = -a
• |x| < a  —->  -a < x < a
• |x| > a  —->  x < -a  or  x > a

If they had seen (and participated in) the development of these steps, why do they struggle? Some students tried to “split” every problem into two, others tried to “trap” the expression between -a and a every time. I started giving memorization tips, and performance has definitely improved. I tell my students to look at the three problems and notice that only one time does the expression get “trapped” or “sandwiched” between -a and a, and that’s the case when an absolute value is less than a positive number. If you can memorize that case, all you have to remember is that the other two cases get split.

Another pointer I give is to create a series of three note cards.

• Front: |x-3| = 7    Back: x-3 = -7  or  x-3 = 7
• Front: |2x+1| < 8    Back: -8 < 2x-1 < 8
• Front: |3x-2| > 10    Back: 3x-2 < -10  or  3x-2 > 10

They can cycle through these cards on a regular basis to quiz themselves on the first step to solve these problems, or as a quick periodic refresher while preparing for the exam.

(Side Note: I go into a long story about how in the “old days” when we had records there was a kind of music called disco. Instead of going to clubs we went to the disco, and the popular pick-up line was “What’s your sign?” When you are working with absolute value equations and inequalities, “What’s your sign?” is the right question to ask, because the sign tells you how to proceed.)

When a is Negative

A similar problem used to occur when a was a negative number. My students would automatically answer “No Solution” for every problem in which a was negative, even though the solution set for |x|>a is the set of all real numbers when a is negative. Now I use the same approach as above.

• |x| = a  —-> No Solution when a is negative
• |x| < a  —->  No Solution when a is negative
• |x| > a  —->  R when a is negative

There is one case that is different than the two others, and if students can remember that exception, then they get the other two cases for free. I do recommend that my students make three note cards for these three scenarios so they can quiz themselves and review these cases whenever they have a few free moments.

Do you have any tips for teaching this material that you’d like to share? Please share your experience and thoughts by leaving a comment, or reaching me through the contact page at my web site – georgewoodbury.com.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article related to General Teaching on my blog. 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  |  August 25, 2010 at 11:39 pm

Since the absolute value is defined piecewise, I wonder if it would be better to ask students to solve absolute value problems by breaking into cases, depending on whether the expression inside the absolute value is positive or negative. One would solve each case separately, and then take the union of the two solution sets.

The disadvantage is that more steps would be required to solve textbook problems. On the other hand, the piecewise approach gives a common methodology for all absolute value problems, and it works in cases where the textbook methods fail (e.g. |x+2| < 2x).

I haven't tried this approach in my own teaching; I'm just thinking out loud.

• 2. zain ahmed  |  October 22, 2011 at 11:11 am

I don’t understand this concept on what you are doing can you do the same thing in another way so i can learn another way or someone who is reading this can you help me on this.