What is an absolute value equation that has the solutions X=6 X=18?

Answers (2)

I never heard of an "absolute value equation". It occurs to me that you might mean a second power equation, also called "parabolic curve".

y = (x - 6)(x - 18)

Looks like this:
www.wolframalpha.com/input/?ab=c&i=plot+y+%3D+(x+-+6)(x+-+18)

You can also express that as |y| = (x - 6)(x - 18) to emphasize the "absolute value" part. (That makes two curves on the plot.)

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A big part of math is pattern recognition. A lot of homework is just fighting with stuff so you will remember the pattern when you see it again. The most common pattern is (x + a) * (x + b) = x^2 + (a + b)x + ab and the special case (x + a) * (x - a) = x^2 - a^2. When you spot the pattern you can just write the answer from memory.

Now study this page until it seems obvious: en.wikipedia.org/wiki/Polynomial_remainder_theorem

I've never seen such a question, so let's walk through the steps.
First, the absolute function generates two points at any nonzero horizontal crossing (it's shaped like a V), so this should be doable with an internal linear function.
Arbitrarily chose x=18 to generate a positive and 6 negative.
What happens if we take 0:
|ax + b| = 0
=> 18 * a + b = 0 ; -(6 * a + b) = 0 => a = 0 ; b = 0
Not a very interesting solution.
Let's try again with a nonzero constant:
|ax + b| = c
=> 18 * a + b = c ; -(6 * a + b) = c
=> 18a + b = -6a - b => b = -12a
This generates an infinite number of equations defined by this relation - basically, multiply both sides by a constant. Let's pick a = 1 for convenience.
a = 1, b = -12, c = 6
|x - 12| = 6
Testing it out:
|6 - 12| = |-6| = 6 ; |18 - 12| = |6| = 6
QED

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