To solve the equation \( |3x - 6| = -|x - 5| + 4 \), we need to break it down into steps:

### Step 1: Consider the two cases of absolute values.

For absolute value equations, we solve for both the positive and negative versions of the expressions inside the absolute value signs.

1. \( |3x - 6| = 3x - 6 \) if \( 3x - 6 \geq 0 \) or \( x \geq 2 \), and \( |3x - 6| = -(3x - 6) \) if \( 3x - 6 < 0 \) or \( x < 2 \).
2. \( |x - 5| = x - 5 \) if \( x \geq 5 \), and \( |x - 5| = -(x - 5) \) if \( x < 5 \).

We will examine both cases based on these conditions.

### Case 1: \( x \geq 5 \)
- In this range, \( |3x - 6| = 3x - 6 \) and \( |x - 5| = x - 5 \).
  
The equation becomes:
\[
3x - 6 = -(x - 5) + 4
\]
Simplifying the right-hand side:
\[
3x - 6 = -x + 5 + 4
\]
\[
3x - 6 = -x + 9
\]
Now, add \( x \) to both sides:
\[
4x - 6 = 9
\]
Add 6 to both sides:
\[
4x = 15
\]
Divide by 4:
\[
x = \frac{15}{4} = 3.75
\]
However, \( x = 3.75 \) is not in the range \( x \geq 5 \). Therefore, there is **no solution** in this case.

### Case 2: \( 2 \leq x < 5 \)
- In this range, \( |3x - 6| = 3x - 6 \) and \( |x - 5| = -(x - 5) \).
  
The equation becomes:
\[
3x - 6 = -(-(x - 5)) + 4
\]
Simplifying:
\[
3x - 6 = x - 5 + 4
\]
\[
3x - 6 = x - 1
\]
Subtract \( x \) from both sides:
\[
2x - 6 = -1
\]
Add 6 to both sides:
\[
2x = 5
\]
Divide by 2:
\[
x = \frac{5}{2} = 2.5
\]
Since \( x = 2.5 \) is within the range \( 2 \leq x < 5 \), this is a valid solution.

### Case 3: \( x < 2 \)
- In this range, \( |3x - 6| = -(3x - 6) \) and \( |x - 5| = -(x - 5) \).
  
The equation becomes:
\[
-(3x - 6) = -(-(x - 5)) + 4
\]
Simplifying:
\[
-3x + 6 = x - 5 + 4
\]
\[
-3x + 6 = x - 1
\]
Add \( 3x \) to both sides:
\[
6 = 4x - 1
\]
Add 1 to both sides:
\[
7 = 4x
\]
Divide by 4:
\[
x = \frac{7}{4} = 1.75
\]
Since \( x = 1.75 \) is in the range \( x < 2 \), this is a valid solution.

### Final Answer:
The two valid solutions are \( x = 2.5 \) and \( x = 1.75 \).

Would you like further clarification or additional details on any step?

Here are 5 related questions to deepen your understanding:
1. How do you solve absolute value inequalities?
2. What are the properties of absolute values in equations?
3. Can an absolute value equation have no solution? Why?
4. How does case analysis help in solving piecewise functions?
5. What happens when absolute values appear in inequalities?

**Tip:** When solving absolute value equations, always check the solution against the conditions of the case.

Learn how to solve the absolute value equation |3x - 6| = -|x - 5| + 4 using case analysis. This step-by-step solution covers key algebraic concepts, including properties of absolute values and case breakdowns. Suitable for students in Grades 8-10.

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