Question

The range of \( x \) which satisfy the inequality \[ -5 \leq \frac{2 - 3x}{4} \leq 9 \] is

• A. \( \left( -\frac{34}{3}, -\frac{22}{3} \right) \)
• B. \( \left[ -\frac{34}{3}, \frac{22}{3} \right] \)
• C. \( \left[ -\frac{34}{3}, 8 \right] \)
• D. \( \left[ -\frac{34}{3}, \frac{22}{3} \right) \)

Hint:

When solving inequalities involving fractions, first eliminate the denominator by multiplying all parts of the inequality by the denominator. Always remember to reverse the inequality when dividing by a negative number.

Answer 1

The Correct Option is D

We are given the inequality: \[ -5 \leq \frac{2 - 3x}{4} \leq 9 \] Step 1: Eliminate the denominator Multiply all parts of the inequality by 4 to eliminate the denominator: \[ 4 \times -5 \leq 2 - 3x \leq 4 \times 9 \] This simplifies to: \[ -20 \leq 2 - 3x \leq 36 \] Step 2: Solve for \( x \) Now, subtract 2 from all parts of the inequality: \[ -20 - 2 \leq -3x \leq 36 - 2 \] This simplifies to: \[ -22 \leq -3x \leq 34 \] Now, divide all parts by -3. Remember, when dividing by a negative number, the direction of the inequality reverses: \[ \frac{-22}{-3} \geq x \geq \frac{34}{-3} \] This simplifies to: \[ \frac{22}{3} \geq x \geq -\frac{34}{3} \] Thus, the range of \( x \) is \( \left[ -\frac{34}{3}, \frac{22}{3} \right) \). Thus, the correct answer is \( \boxed{\left[ -\frac{34}{3}, \frac{22}{3} \right)} \).