Solve the System of Absolute Value Equations: |x-11|-1=5 and |x-5|=0

To solve this system of absolute value equations, we follow these steps:

• Step 1: Solve the equation $|x-5|=0$.
• Step 2: Validate the result with the other equation $||x-11|-1|=5$.

Let's work through each step:

Step 1: Solve $|x-5|=0$.
The equation $|x-5| = 0$ implies that the expression inside the absolute value must be zero. Therefore, $x-5=0$ gives us:

$x = 5$

Step 2: Validate with $||x-11|-1|=5$.
Now, substitute $x = 5$ into the other equation:

$|5-11| = | -6 | = 6$

$|6-1| = |5| = 5$

This simplifies to $|5| = 5$, which is true. Therefore, the solution $x = 5$ satisfies both equations.

Thus, the system of equations has a single solution: $x = 5$.