Does |-x+3| equal x-3?

Question

I'm not sure how to solve absolute values. How do I find the value of an absolute value.

Here are some examples: Does |-x+3| equal x-3? Does |x+3| equal x+3?

Answer 1

we have $$|-x+3|=-x+3$$ if $$3\geq x$$ $$|-x+3|=-(-x)+3)=x-3$$ if $$3<x$$

Answer 2

Nope.

For any function $f$, $|f(x_0)|=f(x_0)$ if and only if $f(x_0)\ge 0$; $|f(x_0)|=-f(x_0)$ if $f(x_0)\lt 0$.

On your particular occasion, doing case work, one could easily find that:

When $-x+3\ge 0\Rightarrow x\le 3, |-x+3|=-x+3$; When $-x+3\lt 0\Rightarrow x\gt 3, |-x+3|=x-3$.

When $x+3\ge 0\Rightarrow x\ge -3, |x+3|=x+3$; When $x+3\lt 0\Rightarrow x\lt -3, |x+3|=-x-3$.