Inequalities with more than one absolute value

Question

I saw a question which asked to find all the solutions to:

$|x+2|+|x-5|=7$

For $x\leq -2$, the answer is $-2$.

For $-2< x <5$, the answer is $R$.

For $x>5$, the answer is $5$.

First I thought the answer should be $(-2, 5)$, but then I realized $-2$ and $5$ solve it too! What would you do?

Answer 1

We have

$$|x+2|+|x-5| = \begin{cases} -(x+2) - (x-5) & \text{ if } x \leq -2 \\ +(x+2) - (x-5) & \text{ if } 2 < x \leq 5 \\ +(x+2) + (x-5) & \text{ if } x > 5 \end{cases} $$

Simplifying,

$$|x+2|+|x-5| = \begin{cases} -2x+3 & \text{ if } x \leq -2 \\ 7 & \text{ if } 2 < x \leq 5 \\ 2x-3 & \text{ if } x > 5 \end{cases} $$

So what does this tell us?