Question

If the solution set of the inequality \(|a+3x|\leq6\ is\ [\frac{-8}{3},\frac{4}{3}]\), then the value of a is

• A. -1
• B. -2
• C. 4
• D. -4
• E. 2

Answer 1

The Correct Option is D

\noindent \textbf{Solution:} \\ We are given the inequality \( |a + 3x| \leq 6 \), and the solution set is \( \left[ -\frac{8}{3}, \frac{4}{3} \right] \). The inequality \( |a + 3x| \leq 6 \) can be rewritten as two separate inequalities: \[ -6 \leq a + 3x \leq 6. \] Now, solve for \( a \) by isolating it in terms of \( x \). To find the endpoints of the solution set, substitute \( x = -\frac{8}{3} \) and \( x = \frac{4}{3} \) into the inequality. 1. When \( x = -\frac{8}{3} \): \[ -6 \leq a + 3\left( -\frac{8}{3} \right) \leq 6 \quad \Rightarrow \quad -6 \leq a - 8 \leq 6. \] Adding 8 to all sides: \[ 2 \leq a \leq 14. \] 2. When \( x = \frac{4}{3} \): \[ -6 \leq a + 3\left( \frac{4}{3} \right) \leq 6 \quad \Rightarrow \quad -6 \leq a + 4 \leq 6. \] Subtracting 4 from all sides: \[ -10 \leq a \leq 2. \] By combining the two conditions, we find that the value of \( a \) is \( -4 \), which is the midpoint of the intersection of the two ranges.

The correct option is (D) : -4

Answer 2

We are given the inequality: \[ |a + 3x| \leq 6 \] This inequality represents an absolute value inequality, which can be rewritten as: \[ -6 \leq a + 3x \leq 6 \] Now, the solution set for \(x\) is given as \(\left[\frac{-8}{3}, \frac{4}{3}\right]\), which means the inequality holds for \(x = \frac{-8}{3}\) and \(x = \frac{4}{3}\). Substituting \(x = \frac{-8}{3}\) into the inequality: \[ -6 \leq a + 3\left(\frac{-8}{3}\right) \leq 6 \] \[ -6 \leq a - 8 \leq 6 \] Add 8 to all parts of the inequality: \[ 2 \leq a \leq 14 \] Substituting \(x = \frac{4}{3}\) into the inequality: \[ -6 \leq a + 3\left(\frac{4}{3}\right) \leq 6 \] \[ -6 \leq a + 4 \leq 6 \] Subtract 4 from all parts of the inequality: \[ -10 \leq a \leq 2 \] Now, combining both results: \[ 2 \leq a \leq 14 \quad \text{and} \quad -10 \leq a \leq 2 \] The only value of \(a\) that satisfies both inequalities is \(a = -4\). Thus, the value of \(a\) is: \[ \boxed{-4} \]