Solve for x: Navigating Nested Absolute Values with the Condition x > 10

To solve the given problem, let's break it down into manageable steps:

The given equation is $||x-11|-1|=5$ and the condition is $x > 10$.

Step 1: Solve the outer absolute value: $||y|-1|=5$, where $y = x - 11$.

This implies two scenarios:

• $|y|-1 = 5 \implies |x-11| = 6$
• $|y|-1 = -5 \implies |x-11| = -4$ (which is not possible since absolute values are non-negative)

So, we only consider $|x-11| = 6$.

Step 2: Solve the equation $|x-11| = 6$, giving two possibilities:

• $x - 11 = 6 \implies x = 17$
• $x - 11 = -6 \implies x = 5$

Step 3: Apply the condition $x > 10$.

Out of the two possible solutions, $x = 5$ does not satisfy $x > 10$.

Therefore, the only valid solution that satisfies both conditions is $x = 17$.

The final solution is $x = 17$.