Let's solve the equation \( 2|x - 1| = 8 + 2x \).

### Step 1: Simplify both sides
Divide both sides of the equation by 2 to make it easier to work with:

\[
|x - 1| = 4 + x
\]

### Step 2: Break into two cases
Since we're dealing with an absolute value equation, we must consider two cases: when the expression inside the absolute value is positive, and when it's negative.

#### Case 1: \( x - 1 \geq 0 \) (i.e., \( x \geq 1 \))
In this case, \( |x - 1| = x - 1 \). The equation becomes:

\[
x - 1 = 4 + x
\]

Subtract \( x \) from both sides:

\[
-1 = 4
\]

This is a contradiction, so there is **no solution** for this case.

#### Case 2: \( x - 1 < 0 \) (i.e., \( x < 1 \))
In this case, \( |x - 1| = -(x - 1) = -x + 1 \). The equation becomes:

\[
-x + 1 = 4 + x
\]

Now, solve for \( x \):

1. Add \( x \) to both sides:
   \[
   1 = 4 + 2x
   \]

2. Subtract 4 from both sides:
   \[
   -3 = 2x
   \]

3. Divide by 2:
   \[
   x = -\frac{3}{2}
   \]

### Final Answer: 
The solution is \( x = -\frac{3}{2} \).

Would you like further clarification on any part of this? Here are some related questions for practice:

1. Solve: \( |x + 2| = 3x + 1 \).
2. Solve: \( 3|x - 4| = 5x + 9 \).
3. Solve: \( |2x + 5| = x - 1 \).
4. Solve: \( 4|x + 1| = x^2 - 3 \).
5. Solve: \( |x - 3| + 2x = 7 \).

**Tip:** When dealing with absolute value equations, always remember to consider both positive and negative cases separately.

Explore how to solve the absolute value equation 2|x - 1| = 8 + 2x with a clear, step-by-step breakdown. Learn about absolute value properties and linear equation solving methods, suitable for students in Grades 8-10.

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