Algebra: Absolute Value Equations
Absolute Value Equations
If you're trying to solve an equation containing a variable that's trapped inside absolute value bars, you'll need to slightly modify your technique. Here's why: Such equations may, in fact, have two answers, instead of just one! That might be initially shocking and exciting (like finding out that while you thought you had one secret admirer, you actually have two), once you figure it all out in the end, everything makes sense. (It must be because you are one irresistible hunk of burnin' love.)
Here's what to do if you encounter an equation whose poor, defenseless variable is trapped in absolute value bars:
- Isolate the absolute value expression. Just like you isolated the variables before, this time isolate that entire expression that falls between the absolute value bars. Follow the same steps as beforestart by adding or subtracting things out of the way and finish by eliminating a coefficient, if the expression has one.
- Create two new equations. Here's the tricky part. You're actually going to design two completely separate equations from the original one. The first equation should look just like the original, just without the bars on it. The second should look just like the first, only take the opposite of the right side of the equation. This might sound tricky, but trust me, it's easy.
- Solve the new equations to get your answer(s). Both of the solutions you get are answers to the original absolute value equation.
You've Got Problems
You're creating two separate equations because absolute values change two different values (any number and its opposite) into the same thing. If you don't understand what I mean, check out the end of
To remind myself that absolute value equations require two separate parts, I sometimes imagine that those absolute value bars are little bars of dynamite that blow that original equation in half, creating two distinct pieces. Now that you get the idea, let me show you how to handle the explosives correctly.
Example 3: Solve the equation 4|2x - 3| + 1 = 21.
Solution: Start by isolating the absolute value quantity on the left. To do so, first subtract 1 from both sides.
- 4|2x - 3| = 20
To complete the isolation process, divide both sides by 4.
- 4|2x - 3|4 = 204
- |2x - 3| = 5
Now that only the absolute values remain on the left side, it's time to create two new equations. The first looks just like the above equation (without bars attached); its sister equation is an exact replica, except its right side will be the opposite of its sibling's right side (it'll have -5 instead of 5).
| 2x - 3 = 5 | 2x - 3 = -5 |
Solve those equations separately.
| 2x - 3 = 5 | 2x - 3 = -5 |
| 2x = 8 | 2x = -2 |
| x = 4 | x = -1 |
You've Got Problems
Problem 4: Solve the equation |x - 5| - 6 = 4.
There you go; the answers are -1 and 4. Do you find two answers hard to swallow? Watch what happens when I check them both in the original equation.
| 4|2(4) - 3| + 1 = 21 | 4|2(-1)-1|+1=21 |
| 4|5| + 1 = 21 | 4|-5| + 1 = 21 |
| 4(5) + 1 = 21 | 4(5) + 1 = 21 |
| 20 + 1 = 21 | 20 + 1 =21 |
Notice that the contents of the absolute values are opposites, so once the absolute value is taken, you end up getting the same results.
Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.
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