Solve `|(x+2)/(x-1)|=2`

Explanation

To solve the equation \( \left| \frac{x+2}{x-1} \right| = 2 \), we will break it down into two cases based on the definition of absolute value.

### Step 1: Set up the two cases
The equation \( \left| \frac{x+2}{x-1} \right| = 2 \) can be split into two separate equations:
1. \( \frac{x+2}{x-1} = 2 \)
2. \( \frac{x+2}{x-1} = -2 \)

### Step 2: Solve the first case
**Case 1:** \( \frac{x+2}{x-1} = 2 \)

To eliminate the fraction, we can cross-multiply:
\[ x + 2 = 2(x - 1) \]
Expanding the right side:
\[ x + 2 = 2x - 2 \]
Now, rearranging the equation to isolate \( x \):
\[ 2 + 2 = 2x - x \]
\[ 4 = x \]
Thus, from Case 1, we have:
\[ x = 4 \]

### Step 3: Solve the second case
**Case 2:** \( \frac{x+2}{x-1} = -2 \)

Again, we will cross-multiply:
\[ x + 2 = -2(x - 1) \]
Expanding the right side:
\[ x + 2 = -2x + 2 \]
Rearranging the equation:
\[ x + 2x = 2 - 2 \]
\[ 3x = 0 \]
Thus, from Case 2, we have:
\[ x = 0 \]

### Step 4: Summary of solutions
The solutions to the equation \( \left| \frac{x+2}{x-1} \right| = 2 \) are:
\[ x = 4 \quad \text{and} \quad x = 0 \]

### Final Answer
The values of \( x \) that satisfy the equation are \( x = 4 \) and \( x = 0 \).

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