Question

The solution of the inequality \( |3x - 4| \leq 5 \) is

• A. \( \left[ -\frac{1}{3}, 3 \right] \)
• B. \( [-1, 4] \)
• C. \( [1, \infty) \)
• D. \( [-1, 1] \)
• E. \( [0, 1] \)

Hint:

For inequalities involving absolute values, isolate the expression inside the absolute value, remove the absolute value, and then solve the resulting compound inequality. Remember to consider both the positive and negative cases.

Answer 1

The Correct Option is A

We are given the inequality: \[ |3x - 4| \leq 5. \] By the definition of absolute value, we know that: \[ |A| \leq B \text{ implies } -B \leq A \leq B. \] Thus, we can rewrite the inequality as: \[ -5 \leq 3x - 4 \leq 5. \] Step 1: Solve the inequality Add 4 to all parts of the inequality to isolate the term with \( x \): \[ -5 + 4 \leq 3x \leq 5 + 4 \] \[ -1 \leq 3x \leq 9. \] Step 2: Divide by 3 to solve for \( x \) \[ \frac{-1}{3} \leq x \leq \frac{9}{3} \] \[ -\frac{1}{3} \leq x \leq 3. \]
Thus, the solution to the inequality is: \[ x \in \left[ -\frac{1}{3}, 3 \right]. \]
Therefore, the correct answer is option (A), \( \left[ -\frac{1}{3}, 3 \right] \).