To solve the equation \(|3-x| + 1 = |2x-2|\), we need to consider different cases based on the definition of absolute value functions. The absolute value function splits into two cases depending on whether the expression inside the absolute value is non-negative or negative.

### Step 1: Define the cases
We have two absolute value expressions, so we need to analyze both:

1. \(3-x \geq 0\) (i.e., \(x \leq 3\)) and \(2x-2 \geq 0\) (i.e., \(x \geq 1\))
2. \(3-x < 0\) (i.e., \(x > 3\)) and \(2x-2 < 0\) (i.e., \(x < 1\))

### Case 1: \(x \leq 3\) and \(x \geq 1\) 
Here, \(3-x \geq 0\) and \(2x-2 \geq 0\). Thus, the equation becomes:

\[
(3-x) + 1 = 2x - 2
\]

Simplify and solve for \(x\):

\[
4 - x = 2x - 2
\]

\[
4 + 2 = 2x + x
\]

\[
6 = 3x
\]

\[
x = 2
\]

Since \(x = 2\) is in the range \(1 \leq x \leq 3\), it is a valid solution for this case.

### Case 2: \(x > 3\)
Here, \(3-x < 0\) and \(2x-2 \geq 0\). Thus, the equation becomes:

\[
-(3-x) + 1 = 2x - 2
\]

Simplify and solve for \(x\):

\[
-x + 3 + 1 = 2x - 2
\]

\[
4 - x = 2x - 2
\]

\[
4 + 2 = 2x + x
\]

\[
6 = 3x
\]

\[
x = 2
\]

But \(x = 2\) does not satisfy \(x > 3\), so there is no solution in this case.

### Case 3: \(x < 1\)
Here, \(3-x \geq 0\) and \(2x-2 < 0\). Thus, the equation becomes:

\[
3-x + 1 = -(2x-2)
\]

Simplify and solve for \(x\):

\[
4 - x = -2x + 2
\]

\[
4 - 2 = -2x + x
\]

\[
2 = -x
\]

\[
x = -2
\]

Since \(x = -2\) is in the range \(x < 1\), it is a valid solution for this case.

### Final Solution:
The solutions to the equation \(|3-x| + 1 = |2x-2|\) are \(x = 2\) and \(x = -2\).

Would you like a detailed explanation of any step, or do you have any questions?

Here are five related questions:
1. How do you solve equations involving more complex absolute value expressions?
2. What are the geometric interpretations of absolute value equations?
3. How do you check if a solution falls within a given range?
4. How do absolute value inequalities differ from equations?
5. What happens when there are more than two absolute value expressions in an equation?

**Tip:** When solving absolute value equations, always check each solution in the original equation to ensure it satisfies the initial conditions.

Explore how to solve the absolute value equation |3-x| + 1 = |2x-2| with detailed step-by-step solutions. This problem demonstrates key algebraic concepts including piecewise functions and absolute value properties, suitable for students in Grades 9-11.

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