Equation with absolute value

Solve this equation over the real domain: $$||x-2|-3|=5$$.

• Resolution

• $$x\le 2: |-x+2-3|=5 \longrightarrow |-x-1|=5 \longrightarrow |-x-1|=5$$
1. $$x\le -1: -x-1=5 \longrightarrow x=-6$$
2. $$x\ge -1: x+1=5 \longrightarrow x=4$$ has no solution
• $$x\ge 2: |x-2-3|=5 \longrightarrow |x-5|=5$$
1. $$x\le 5: -x+5=5 \longrightarrow x=0$$ has no solution
2. $$x\ge -1: x-5=5 \longrightarrow x=10$$
• Result

The equation has two solutions: $$x=-6$$ and $$x=10$$.