Question

If \( x \) satisfies the inequality \( -3<\frac{1}{2} + \frac{-3x}{2} \leq 6 \), then \( x \) lies in the interval

• A. \( \left[ \frac{-11}{3}, \frac{7}{3} \right) \)
• B. \( \left( \frac{-11}{3}, \frac{7}{3} \right] \)
• C. \( \left[ \frac{7}{3}, \frac{11}{3} \right] \)
• D. \( \left[ \frac{-10}{3}, \frac{7}{3} \right] \)
• E. \( \left[ \frac{7}{3}, \frac{10}{3} \right] \)

Hint:

When solving inequalities, be careful when multiplying or dividing by negative numbers as it flips the inequality sign.

Answer 1

The Correct Option is A

The inequality is: \[ -3<\frac{1}{2} + \frac{-3x}{2} \leq 6 \] Subtract \( \frac{1}{2} \) from all sides: \[ -3 - \frac{1}{2}<\frac{-3x}{2} \leq 6 - \frac{1}{2} \] Simplifying the terms: \[ -\frac{7}{2}<\frac{-3x}{2} \leq \frac{11}{2} \] Multiply both sides by \( -2 \) (note that the inequality sign flips when multiplying by a negative number): \[ 7<3x \leq -11 \] Finally, divide by 3: \[ \frac{7}{3}<x \leq \frac{-11}{3} \] Thus, the solution is \( x \in \left[ \frac{-11}{3}, \frac{7}{3} \right] \). Thus, the correct answer is (A).