https://file.aiquickdraw.com/mathbot/file/1725374818546iey4xx5a.jpg

The equation shown in the image is:

\[
|-5x + 8| = |2x - 2|
\]

To solve this absolute value equation, we'll consider the cases where the expressions inside the absolute values are either equal or their negatives are equal.

### Case 1: 
\[
-5x + 8 = 2x - 2
\]

**Solve for \(x\):**

Add \(5x\) to both sides:

\[
8 = 7x - 2
\]

Add 2 to both sides:

\[
10 = 7x
\]

Divide by 7:

\[
x = \frac{10}{7}
\]

### Case 2: 
\[
-5x + 8 = -(2x - 2)
\]

Simplify the right-hand side:

\[
-5x + 8 = -2x + 2
\]

**Solve for \(x\):**

Add \(5x\) to both sides:

\[
8 = 3x + 2
\]

Subtract 2 from both sides:

\[
6 = 3x
\]

Divide by 3:

\[
x = 2
\]

### Solutions
The solutions to the equation are \( x = \frac{10}{7} \) and \( x = 2 \).

Would you like more details on these steps or have any further questions?

Here are some related questions for practice:

1. How would you solve an absolute value equation with three terms, such as \(|ax + b| + |cx + d| = |ex + f|\)?
2. What happens when the expressions inside the absolute values are more complex, like quadratic expressions?
3. How do you solve an absolute value inequality, such as \(|ax + b| < c\)?
4. What are the key differences between solving absolute value equations and inequalities?
5. How would you graph the equation \(|-5x + 8| = |2x - 2|\)?

**Tip:** When solving absolute value equations, always remember to check your solutions in the original equation, as extraneous solutions can sometimes appear.

Learn how to solve the absolute value equation |-5x + 8| = |2x - 2| with a detailed step-by-step solution. This problem involves algebraic concepts such as absolute values and solving linear equations, suitable for students in Grades 9-10.

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