If `|x + 1| = |3x - 2|` what are the possible values of x?

Options

A. `1/4 and 3/4`
B. `1/4 and 3/2`
C. `2/3 and 3/2`
D. `2/3 and 4/3`

Answer

To solve the equation \( |x + 1| = |3x - 2| \), we need to consider the different cases that arise from the absolute value expressions.

### Step 1: Set Up Cases
The absolute value equation can be solved by considering the following four cases:

1. **Case 1:** Both expressions are positive:
\[ x + 1 = 3x - 2 \]

2. **Case 2:** The left expression is positive, and the right expression is negative:
\[ x + 1 = -(3x - 2) \]

3. **Case 3:** The left expression is negative, and the right expression is positive:
\[ -(x + 1) = 3x - 2 \]

4. **Case 4:** Both expressions are negative:
\[ -(x + 1) = -(3x - 2) \]

### Step 2: Solve Each Case

**Case 1:**
\[ x + 1 = 3x - 2 \]
Rearranging gives:
\[ 1 + 2 = 3x - x \implies 2 = 2x \implies x = 1 \]

**Case 2:**
\[ x + 1 = - (3x - 2) \]
This simplifies to:
\[ x + 1 = -3x + 2 \]
Rearranging gives:
\[ x + 3x = 2 - 1 \implies 4x = 1 \implies x = \frac{1}{4} \]

**Case 3:**
\[ -(x + 1) = 3x - 2 \]
This simplifies to:
\[ -x - 1 = 3x - 2 \]
Rearranging gives:
\[ -x - 3x = -2 + 1 \implies -4x = -1 \implies x = \frac{1}{4} \]

**Case 4:**
\[ -(x + 1) = -(3x - 2) \]
This simplifies to:
\[ -x - 1 = -3x + 2 \]
Rearranging gives:
\[ -x + 3x = 2 + 1 \implies 2x = 3 \implies x = \frac{3}{2} \]

### Step 3: Collect Solutions
From the four cases, we have found the following possible values for \( x \):
- From Case 1: \( x = 1 \)
- From Case 2: \( x = \frac{1}{4} \)
- From Case 3: \( x = \frac{1}{4} \) (duplicate)
- From Case 4: \( x = \frac{3}{2} \)

Thus, the unique solutions are:
\[ x = \frac{1}{4} \quad \text{and} \quad x = \frac{3}{2} \]

### Final Answer
The possible values of \( x \) are:
\[ \frac{1}{4} \quad \text{and} \quad \frac{3}{2} \]