Solution - Absolute value equations

Use the rules:
$$|x|=|y|$$ → $$x=\pm y$$ and $$|x|=|y|$$ → $$\pm x=y$$
to write all four options of the equation
$$|x|=-|x-9|$$
without the absolute value bars:

When simplified, equations $$x=+y$$ and $$+x=y$$ are the same and equations $$x=-y$$ and $$-x=y$$ are the same, so we end up with only 2 equations:

The equation is false so it has no solution.

$$x=\frac{9}{2}$$
(1 solution(s))

Each line represents the function of one side of the equation:
$$y=|x|$$
$$y=-|x-9|$$
The equation is true where the two lines cross.