The value of x if `0 le |2x + 3| le 3` belongs to

Options

A. [-3, 0]
B. [1, 2]
C. [2, 3]
D. [-2, -1]

Answer

To solve the inequality \(0 \leq |2x + 3| \leq 3\), we will break it down into two parts:

1. \( |2x + 3| \geq 0 \)
2. \( |2x + 3| \leq 3 \)

### Step 1: Analyze the first part \( |2x + 3| \geq 0 \)

The absolute value of any expression is always non-negative. Therefore, this part is always true for all \(x\). We can focus on the second part.

### Step 2: Solve the second part \( |2x + 3| \leq 3 \)

The absolute value inequality \( |A| \leq B \) can be rewritten as:
\[ -B \leq A \leq B \]
In our case, \(A = 2x + 3\) and \(B = 3\). Thus, we have:
\[ -3 \leq 2x + 3 \leq 3 \]

### Step 3: Break it into two inequalities

We will break this compound inequality into two separate inequalities:

1. \( -3 \leq 2x + 3 \)
2. \( 2x + 3 \leq 3 \)

### Step 4: Solve the first inequality \( -3 \leq 2x + 3 \)

Subtract 3 from both sides:
\[ -3 - 3 \leq 2x \]
\[ -6 \leq 2x \]
Now, divide by 2:
\[ -3 \leq x \]

### Step 5: Solve the second inequality \( 2x + 3 \leq 3 \)

Subtract 3 from both sides:
\[ 2x \leq 3 - 3 \]
\[ 2x \leq 0 \]
Now, divide by 2:
\[ x \leq 0 \]

### Step 6: Combine the results

From the two inequalities we derived:
1. \( x \geq -3 \)
2. \( x \leq 0 \)

Combining these results, we find:
\[ -3 \leq x \leq 0 \]

### Final Result

Thus, the values of \(x\) that satisfy the original inequality \(0 \leq |2x + 3| \leq 3\) belong to the interval:
\[ x \in [-3, 0] \]