Question

The correct solution of \(-22 < 8x - 6 \leq 26\) is the interval:

• A. \((-2, 4]\)
• B. \([-2, 4]\)
• C. \((-2, 4)\)
• D. \([-2, 4)\)

Answer 1

The Correct Option is A

To solve the compound inequality \(-22 < 8x - 6 \leq 26\), we will break it into two separate inequalities and solve each one individually.

1. Solve the first inequality: \(-22 < 8x - 6\) 

Add 6 to both sides:

\(-22 + 6 < 8x\)

\(-16 < 8x\)

Divide both sides by 8:

\(-2 < x\)

1. Solve the second inequality: \(8x - 6 \leq 26\)

Add 6 to both sides:

\(8x - 6 + 6 \leq 26 + 6\)

\(8x \leq 32\)

Divide both sides by 8:

\(x \leq 4\)

We combine the solutions from both parts: \(-2 < x \leq 4\)

This represents the interval \((-2, 4]\), which is the correct solution.

Answer 2

The given inequality is:
$-22 < 8x - 6 \le 26$.
Step 1: Break the inequality into two parts.
$-22 < 8x - 6$ and $8x - 6 \le 26$.
Step 2: Solve each part.
1. For $-22 < 8x - 6$:
$-22 + 6 < 8x \implies -16 < 8x \implies x > -2$.
2. For $8x - 6 \le 26$:
$8x \le 26 + 6 \implies 8x \le 32 \implies x \le 4$.
Step 3: Combine the two parts.
The combined inequality is:
$-2 < x \le 4$.
In interval notation:
$x \in (-2, 4]$.
Final Answer:
$(-2, 4]$