Question

The solution set of the inequality \(-2\leq\frac{3x+2}{2}\lt7\) is

• A. {x:3 ≤ x < 4}
• B. {x:-2 ≤ x < 3}
• C. {x:-2 ≤ x < 4}
• D. {x:0 ≤ x < 6}
• E. {x:-2 ≤ x < 6}

Answer 1

The Correct Option is C

We are given the inequality:

\[ -2 \leq \frac{3x + 2}{2} < 7 \]

Step 1: Solve the inequality

We need to solve the compound inequality \( -2 \leq \frac{3x + 2}{2} < 7 \).

First part: \( -2 \leq \frac{3x + 2}{2} \)

Multiply both sides of the inequality by 2 to eliminate the denominator: \[ -4 \leq 3x + 2 \] Now subtract 2 from both sides: \[ -6 \leq 3x \] Finally, divide both sides by 3: \[ -2 \leq x \]

Second part: \( \frac{3x + 2}{2} < 7 \)

Multiply both sides of the inequality by 2 to eliminate the denominator: \[ 3x + 2 < 14 \] Now subtract 2 from both sides: \[ 3x < 12 \] Finally, divide both sides by 3: \[ x < 4 \]

Step 2: Combine the results

From the first part, we found that \( x \geq -2 \), and from the second part, we found that \( x < 4 \). Thus, the solution set is: \[ -2 \leq x < 4 \]

The correct option is (C) : \(\{x:-2 ≤ x < 4\}\)

Answer 2

We are given the inequality: \[ -2 \leq \frac{3x+2}{2} < 7 \] To solve for \(x\), we will break the inequality into two parts: 1. **Solve \( -2 \leq \frac{3x+2}{2} \):** Multiply both sides of the inequality by 2 to eliminate the denominator: \[ -4 \leq 3x + 2 \] Now subtract 2 from both sides: \[ -6 \leq 3x \] Next, divide by 3: \[ -2 \leq x \] 2. **Solve \( \frac{3x+2}{2} < 7 \):** Again, multiply both sides of the inequality by 2: \[ 3x + 2 < 14 \] Now subtract 2 from both sides: \[ 3x < 12 \] Next, divide by 3: \[ x < 4 \] Thus, the solution set is the intersection of the two parts: \[ -2 \leq x < 4 \] So, the solution set is: \[ \boxed{x: -2 \leq x < 4} \]