The solution of `|2x-3| lt |x+2|` is

Options

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

Explanation

To solve the inequality \( |2x - 3| < |x + 2| \), we will break it down step by step.

### Step 1: Understanding the Absolute Value Inequality
The inequality \( |A| < |B| \) implies that \( A \) is less than \( B \) and also \( A \) is greater than \(-B\). Thus, we can write:
1. \( 2x - 3 < x + 2 \)
2. \( 2x - 3 > -(x + 2) \)

### Step 2: Solve the First Inequality
Let's solve the first inequality:
\[ 2x - 3 < x + 2 \]
Subtract \( x \) from both sides:
\[ 2x - x - 3 < 2 \]
This simplifies to:
\[ x - 3 < 2 \]
Now, add 3 to both sides:
\[ x < 5 \]

### Step 3: Solve the Second Inequality
Now, let's solve the second inequality:
\[ 2x - 3 > -(x + 2) \]
Distributing the negative sign gives:
\[ 2x - 3 > -x - 2 \]
Now, add \( x \) to both sides:
\[ 2x + x - 3 > -2 \]
This simplifies to:
\[ 3x - 3 > -2 \]
Now, add 3 to both sides:
\[ 3x > 1 \]
Finally, divide by 3:
\[ x > \frac{1}{3} \]

### Step 4: Combine the Results
Now we have two inequalities:
1. \( x < 5 \)
2. \( x > \frac{1}{3} \)

Combining these gives us:
\[ \frac{1}{3} < x < 5 \]

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