Which of the following is the solution set of `|2x - 3| lt 7`

Options

A. `{x : -5 lt x lt 2}`
B. `{x : -5 lt x lt 5}`
C. `{x : -2 lt x lt 5}`
D. `{x : x lt -5 "or " x gt 2}`

Answer

To solve the inequality \( |2x - 3| < 7 \), we can follow these steps:

### Step 1: Rewrite the Absolute Value Inequality
The absolute value inequality \( |A| < B \) can be rewritten as two inequalities:
\[ -B < A < B \]
In our case, we have:
\[ -7 < 2x - 3 < 7 \]

### Step 2: Split into Two Inequalities
Now we can split this into two separate inequalities:
1. \( 2x - 3 < 7 \)
2. \( 2x - 3 > -7 \)

### Step 3: Solve the First Inequality
Let's solve the first inequality \( 2x - 3 < 7 \):
1. Add 3 to both sides:
\[ 2x < 7 + 3 \]
\[ 2x < 10 \]
2. Divide both sides by 2:
\[ x < 5 \]

### Step 4: Solve the Second Inequality
Now, let's solve the second inequality \( 2x - 3 > -7 \):
1. Add 3 to both sides:
\[ 2x > -7 + 3 \]
\[ 2x > -4 \]
2. Divide both sides by 2:
\[ x > -2 \]

### Step 5: Combine the Results
Now, we combine the results from both inequalities:
\[ -2 < x < 5 \]

### Final Solution Set
The solution set is:
\[ (-2, 5) \]