We are given the equation:

\[
1 - |10 - 5x| = -5
\]

### Step 1: Isolate the absolute value expression
We need to first isolate the absolute value term \(|10 - 5x|\). Subtract 1 from both sides of the equation:

\[
1 - |10 - 5x| - 1 = -5 - 1
\]

\[
-|10 - 5x| = -6
\]

### Step 2: Remove the negative sign
Multiply both sides of the equation by \(-1\):

\[
|10 - 5x| = 6
\]

### Step 3: Solve the absolute value equation
An absolute value equation \(|A| = B\) splits into two cases:

1. \(10 - 5x = 6\)
2. \(10 - 5x = -6\)

### Case 1: \(10 - 5x = 6\)

Subtract 10 from both sides:

\[
-5x = 6 - 10
\]

\[
-5x = -4
\]

Divide by \(-5\):

\[
x = \frac{-4}{-5} = \frac{4}{5}
\]

### Case 2: \(10 - 5x = -6\)

Subtract 10 from both sides:

\[
-5x = -6 - 10
\]

\[
-5x = -16
\]

Divide by \(-5\):

\[
x = \frac{-16}{-5} = \frac{16}{5}
\]

### Final solution:
The two solutions are:

\[
x = \frac{4}{5} \quad \text{or} \quad x = \frac{16}{5}
\]

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you solve absolute value inequalities?
2. Can you graph the solutions to this equation?
3. How do you solve more complex absolute value equations?
4. What happens when there is no solution to an absolute value equation?
5. How do absolute value equations apply to real-world problems?

**Tip:** Always check if isolating the absolute value term simplifies the process when solving such equations.

Learn how to solve the absolute value equation 1 - |10 - 5x| = -5 with a detailed step-by-step explanation. This solution explores key algebra concepts, including absolute value equations and solving linear equations. Suitable for students in Grades 8-10.

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