`|2x - 3| lt |x+5|`, then x belongs to

Options

A. (-3, 5)
B. (5, 9)
C. `(-(2)/(3), 8)`
D. `(-8, (2)/(3))`

Answer

To solve the inequality \( |2x - 3| < |x + 5| \), we will follow these steps:

### Step 1: Understand the Absolute Value Inequality
The inequality \( |A| < |B| \) can be interpreted as:
- \( -B < A < B \)

Thus, we can break down our inequality into two cases:
1. \( -|x + 5| < 2x - 3 < |x + 5| \)

### Step 2: Solve the Two Cases
We will solve the two inequalities separately.

#### Case 1: \( 2x - 3 < x + 5 \)
1. Rearranging gives:
\[ 2x - x < 5 + 3 \]
\[ x < 8 \]

#### Case 2: \( 2x - 3 > - (x + 5) \)
1. Rearranging gives:
\[ 2x - 3 > -x - 5 \]
\[ 2x + x > -5 + 3 \]
\[ 3x > -2 \]
\[ x > -\frac{2}{3} \]

### Step 3: Combine the Results
From the two cases, we have:
1. \( x < 8 \)
2. \( x > -\frac{2}{3} \)

Thus, combining these results, we get:
\[ -\frac{2}{3} < x < 8 \]

### Final Answer
Therefore, the solution to the inequality \( |2x - 3| < |x + 5| \) is:
\[ x \in \left(-\frac{2}{3}, 8\right) \]