Equations with Absolute Value

Date: 01/24/99 at 21:34:51
From: Angela 
Subject: Equations with absolute value or squared quantities

I have problems finding two answers for equations with absolute values. 
I don't know how to start solving the problem. Here's an example:

   | 2k - 5 | = 13

Since you can't just subtract out of the absolute value signs, I divide 
by 5 on both sides of the equation:

   | 2k - 5 | = 13
   ----------   --
        5        5

The fives cancel one another out, and 13/5 = 2 3/5.

So now my problem says:

   | 2k | = 2 3/5

Do I now try to figure out the answer by putting:

   2k = -2 3/5

and also the positive equation?

Date: 01/25/99 at 10:04:23
From: Doctor Rick
Subject: Re: Equations with absolute value or squared quantities

Hi, Angela.

It's true that you can't just subtract from an absolute value, but you 
can divide. The problem is that dividing doesn't help you. Forget the 
absolute value for a moment: (2k - 5)/5 does NOT equal 2k; it equals 
(2/5)k - 1. In order to "undo" (cancel) the addition of -5, you MUST 
SUBTRACT -5 (add 5). You can't cancel an addition by dividing, or 
cancel a multiplication by subtracting.

So, how can you solve the equation? You have to change the absolute 
value equation into two equations, remembering the definition of 
absolute value:

  |x| =  x if x >= 0
        -x if x < 0

If 2k - 5 >= 0 then your equation is the same as

  2k - 5 = 13

because the absolute value has no effect on a non-negative quantity. 
You can solve this:

  2k - 5 + 5 = 13 + 5
          2k = 18
      2k / 2 = 18 / 2
           k = 9

Now (here is the tricky part) you have to go back and see if the 
requirement that 2k - 5 >= 0 has been satisfied. Plug in 9 for k:

  2 * 9 - 5 = 13 >= 0

so yes, the condition is satisfied, and k = 9 is a solution.

We still need to try the other possibility: if 2k - 5 < 0. Then your
equation becomes

  -(2k - 5) = 13

because the absolute value of a negative number is the negative of the
number (to make it positive). Solve this equation:

      -2k + 5 = 13   (distributing the minus sign)
  -2k + 5 - 5 = 13 - 5
          -2k = 8
   -2k / (-2) = 8 / (-2)
            k = -4

Again, we must check to see whether this solution meets our condition: 
2k - 5 < 0. Plug in -4 for k:

  2(-4) - 5 = -13 < 0

Yes, indeed. So we found two solutions to the equation:

  |2k - 5| = 13

  solution: k = 9 or k = -4

Check both solutions by plugging them into the equation, and you're 
done.

Now try another equation and see if you understand the method.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/