https://file.aiquickdraw.com/mathbot/file/1735945928589g5iap9os.jpg

The problem in the image is:

\[
|x - 1| = 7x - 13
\]

### Solution:
To solve this equation, we consider two cases because the absolute value expression can be either positive or negative.

#### Case 1: \( x - 1 \geq 0 \)
If \( x - 1 \geq 0 \), the absolute value expression becomes:
\[
x - 1 = 7x - 13
\]

Now, solve for \( x \):
\[
x - 7x = -13 + 1
\]
\[
-6x = -12
\]
\[
x = 2
\]

Next, check whether \( x = 2 \) satisfies the condition \( x - 1 \geq 0 \):
\[
2 - 1 \geq 0
\]
This is true, so \( x = 2 \) is a valid solution for this case.

#### Case 2: \( x - 1 < 0 \)
If \( x - 1 < 0 \), the absolute value expression becomes:
\[
-(x - 1) = 7x - 13
\]
Simplify:
\[
- x + 1 = 7x - 13
\]

Solve for \( x \):
\[
1 + 13 = 7x + x
\]
\[
14 = 8x
\]
\[
x = \frac{14}{8} = \frac{7}{4}
\]

Now, check whether \( x = \frac{7}{4} \) satisfies the condition \( x - 1 < 0 \):
\[
\frac{7}{4} - 1 < 0
\]
\[
\frac{3}{4} > 0
\]
This is **not true**, so \( x = \frac{7}{4} \) is not a valid solution.

### Final Answer:
The only solution is:
\[
\boxed{x = 2}
\]

Would you like further clarification or a more detailed explanation?

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### Related Questions:
1. What are the steps to solve equations involving absolute values?
2. How do we interpret solutions that do not satisfy initial conditions?
3. Can an absolute value equation have no solutions? If so, under what conditions?
4. What happens when an absolute value equation involves quadratic expressions?
5. How do you graph an absolute value equation like this?

### Tip:
When solving absolute value equations, always check the solutions against the conditions derived from splitting the absolute value into cases!

Learn how to solve the absolute value equation |x - 1| = 7x - 13 step by step. This problem covers absolute value concepts, properties, and case-by-case analysis. Suitable for students in Grades 8-10.

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