I'm a bit confused on how to use algebra to solve absolute value equations.

For example: If |x^2 − 12| = x, which of the following could be the value of x?

A. –4

B. –3

C. 1

D. 2

E. 3

I solved this easily using substitution, but I want to know how to solve it using algebra. I saw a similar problem and tried to copy over the steps but apparently it doesn't work algebraically here. Not sure what I am missing.

From my understanding you assume what's in the absolute value to be positive and negative, then solve for x. Assuming If |x^2 − 12| is positive, we get

x^2 - 12 = +x

x^2 -x -12 = 0

(x-4)(x+3)

x=4 or -3

Then do we just eliminate x=-3 since x has to be positive?

Next, assuming x is negative

x^2 - 12 = -x

x^2 + x - 12 = 0

(x+4)(x-3)

x=-4 or +3

Here, do we just eliminate x=+3 since we assumed x has to be negative?

Or should we not have done this step all together since we cannot assume x to be negative since |x^2 − 12| will never be a negative value?

In which case, how do we get the correct answer of E. x = 3.

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