Solving the Nested Absolute Value Equation: Resolve ||2-7x|-5|=7

To solve this problem, we'll follow these steps:

• Step 1: Consider the equation $|B| = 7$, where $B = |2-7x|-5$.
• Step 2: Solve the two separate scenarios: $B = 7$ and $B = -7$.

Now, let's work through each major step:

Step 1: Begin with $|2-7x|-5 = 7$.
This implies either $|2-7x|-5 = 7$ or $|2-7x|-5 = -7$.

Step 2: Solve $|2-7x|-5 = 7$:

- $|2-7x|-5 = 7$ becomes $|2-7x| = 12$.

- For $|2-7x| = 12$, solve $2-7x = 12$ or $2-7x = -12$.

Step 3: Solve the equations derived:

• Case 1:

- Solve $2-7x = 12$, leading to:

+ $(-7x = 10)$, hence $x = -\frac{10}{7}$.
• Case 2:

- Solve $2-7x = -12$, arrives at:

+ $(-7x = -14)$, so $x = 2$.

Check for the case $|2-7x|-5 = -7$ is invalidated since no real solution exists for absolute difference less than zero.

Conclusively, solving $||2-7x|-5|=7$ yields solutions $x=2$ and $x=-\frac{10}{7}$.

Therefore, the correct choice is "a and c correct".