1. Stating the problem: Solve the equation $$|x - |x|| = 10$$. 2. Understand the absolute value function: $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$. 3. Consider two cases based on the value of $x$: - Case 1: $x \geq 0$ Then, $|x| = x$, so the expression inside the absolute value becomes: $$x - |x| = x - x = 0$$ Thus, $$|x - |x|| = |0| = 0 \neq 10$$, so no solution here. - Case 2: $x < 0$ Then, $|x| = -x$, so the expression inside the absolute value becomes: $$x - |x| = x - (-x) = x + x = 2x$$ The equation becomes: $$|2x| = 10$$ 4. Solve for $x$ in Case 2: $$|2x| = 10 \implies 2|x| = 10 \implies |x| = 5$$ Since $x < 0$, we have: $$x = -5$$ 5. Verify the solution: Substitute $x = -5$ back into the original equation: $$|x - |x|| = |-5 - |-5|| = |-5 - 5| = |-10| = 10$$ This satisfies the equation. Final answer: $$x = -5$$