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Absolute value is one of the simplest functions—and paradoxically, one of the most problematic.
On the face of it, nothing could be simpler: it just means “whatever comes in, a positive number comes out.”
Absolute values seem to give us permission to ignore the whole nasty world of negative numbers and return to the second grade when all numbers were positive.
But consider these three equations. They look very similar—only the number changes—but the solutions are completely different.
works. | doesn’t work. | is the only solution. |
Hey, so does ! | Neither does . | |
Concisely, . | Hey... absolute values are never negative! |
We see that the first problem has two solutions , the second problem has no solutions , and the third problem has one solution . This gives you an example of how things can get confusing with absolute values—and how you can solve things if you think more easily than with memorized rules .
For more complicated problems, follow a three-step approach.
In my experience, most problems with this type of equation do not occur in the first and third step. And they do not occur because students try to think it through (second step) and don’t think it through correctly. They occur because students try to take “shortcuts” to avoid the second step entirely.
So this problem has two answers: and
The “think” step in the above examples was relatively straightforward, because there were no variables on the right side of the equation. When there are variables on the right side, you temporarily “pretend” that the right side of the equation is a positive number, and break the equation up accordingly. However, there is a price to be paid for this slight of hand: you have to check your answers, because they may not work even if you do your math correctly.
We begin by approaching this in analogy to the first problem above, . We saw that could be either 10, or –10. So we will assume in this case that can be either , or the negative of that, and solve both equations.
So we have two solutions: and . Do they both work? Let’s try them both.
We see in this case that the first solution, , worked; the second, , did not. So the only solution to this problem is .
However, there was no way of knowing that in advance. For such problems, the only approach is to solve them twice, and then test both answers . In some cases, both will work; in some cases, neither will work. In some cases, as in this one, one will work and the other will not.
OK, why is that? Why can you do all the math right and still get a wrong answer?
Remember that the problem has two solutions, and has none. We started with the problem . OK, which is that like? Is the right side of the equation like 10 or –10? If you think about it, you can convince yourself that it depends on what is . After you solve, you may wind up with an -value that makes the right side positive; that will work. Or, you may wind up with an -value that makes the right side negative; that won’t work. But you can’t know until you get there.
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