find the solution set of the following equation with absolute value

Question

WA online me results $x = -5$, failed to reach it

$$\left ( \left | x \right |+2 \right )\left ( \left| x-2 \right| -3\right)=x^{2}+3$$

Answer 1

You can break into cases.

Case 1: $x\leq 0$. In this case, $|x|=-x$ and $|x-2|=-x+2$

Case 2: $0\leq x\leq 2$. In this case, $|x|=x$ and $|x-2|=-x+2$

Case 3: $2\leq x$. In this case, $|x|=x$ and $|x-2|=x-2$

For each case, you can try to solve the equality making the necessary replacements, looking only for solutions that match the case you are in.

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For example, in case $3$, you have

$(x+2)(x-2-3)=x^2+3$ which simplifies to

$x^2-5x+2x-10=x^2+3$

$3x=-13$

$x=\frac{-13}{3}$

However, since we were trying to work in the case that $2\leq x$, we have that $\frac{-13}{3}$ is not a valid solution, and so there are no solutions to our original equation with $x\geq 2$.

Continuing through the other cases, you will find the result.