Solve the equation with a nested absolute value: the absolute value of (the absolute value of x, minus 4) equals 5.

Step-by-step solution

1. 1

   Peel the outer bars first. For any expression $A$, the statement $|A|=5$ means $A=5$ or $A=-5$. Treat $A=|x|-4$ as a single block for now:

   $|x|-4=5\qquad\text{or}\qquad |x|-4=-5.$

   Working inside-out instead of outside-in is what makes nested moduli confusing.

2. 2

   Solve the first branch.

   $|x|-4=5\ \Longrightarrow\ |x|=9\ \Longrightarrow\ x=9\ \text{or}\ x=-9.$

3. 3

   Solve the second branch.

   $|x|-4=-5\ \Longrightarrow\ |x|=-1.$

   An absolute value is a distance from zero and can never be negative, so this branch contributes no solutions. Recognising a dead branch early saves the trouble of splitting it further.

4. 4

   Collect the surviving solutions.

   $x=9\quad\text{or}\quad x=-9.$

5. 5

   Verify both in the original equation. At $x=9$: $\bigl||9|-4\bigr|=|9-4|=|5|=5$ ✓. At $x=-9$: $\bigl||-9|-4\bigr|=|9-4|=5$ ✓. Both check out, so the solution set is $\{-9,9\}$.