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.