The solution set of the inequation `|2x - 3| < |x+2|,` is

Options

A. `(-oo, 1//3)`
B. `(1//3, 5)`
C. `(5, oo)`
D. `(-oo, 1//3)cup(5, oo)`

Answer

To solve the inequality \( |2x - 3| < |x + 2| \), we can break it down into different cases based on the critical points where the expressions inside the absolute values change sign.

### Step-by-Step Solution:

1. **Identify Critical Points**:
The critical points occur when the expressions inside the absolute values equal zero:
- \( 2x - 3 = 0 \) gives \( x = \frac{3}{2} \)
- \( x + 2 = 0 \) gives \( x = -2 \)

Thus, the critical points are \( x = -2 \) and \( x = \frac{3}{2} \).

2. **Divide the Number Line**:
We will analyze the inequality in the intervals defined by these critical points:
- \( (-\infty, -2) \)
- \( [-2, \frac{3}{2}) \)
- \( [\frac{3}{2}, \infty) \)

3. **Case 1: \( x < -2 \)**:
In this interval, both expressions inside the absolute values are negative:
\[ |2x - 3| = -(2x - 3) = -2x + 3 \]
\[ |x + 2| = -(x + 2) = -x - 2 \]
The inequality becomes:
\[ -2x + 3 < -x - 2 \]
Simplifying this:
\[ -2x + x < -2 - 3 \]
\[ -x < -5 \quad \Rightarrow \quad x > 5 \]
However, this contradicts our assumption that \( x < -2 \). Therefore, there are no solutions in this interval.

4. **Case 2: \( -2 \leq x < \frac{3}{2} \)**:
In this interval, \( 2x - 3 < 0 \) and \( x + 2 \geq 0 \):
\[ |2x - 3| = -2x + 3 \]
\[ |x + 2| = x + 2 \]
The inequality becomes:
\[ -2x + 3 < x + 2 \]
Simplifying this:
\[ -2x - x < 2 - 3 \]
\[ -3x < -1 \quad \Rightarrow \quad x > \frac{1}{3} \]
Thus, the solution in this interval is:
\[ \frac{1}{3} < x < \frac{3}{2} \]

5. **Case 3: \( x \geq \frac{3}{2} \)**:
In this interval, both expressions inside the absolute values are positive:
\[ |2x - 3| = 2x - 3 \]
\[ |x + 2| = x + 2 \]
The inequality becomes:
\[ 2x - 3 < x + 2 \]
Simplifying this:
\[ 2x - x < 2 + 3 \]
\[ x < 5 \]
Thus, the solution in this interval is:
\[ \frac{3}{2} \leq x < 5 \]

6. **Combine Solutions**:
Now we combine the solutions from the second and third cases:
- From Case 2: \( \frac{1}{3} < x < \frac{3}{2} \)
- From Case 3: \( \frac{3}{2} \leq x < 5 \)

Therefore, the complete solution set is:
\[ \frac{1}{3} < x < 5 \]

### Final Answer:
The solution set of the inequation \( |2x - 3| < |x + 2| \) is:
\[ \boxed{\left( \frac{1}{3}, 5 \right)} \]