Sure, let's solve the inequality step by step. We're given the compound inequality:
[tex]\[ -68 < 8x - 4 < -36 \][/tex]
We'll work on each part of the inequality separately.
### 1. Isolate [tex]\( x \)[/tex] on one side of the inequalities:
#### Part A: [tex]\(-68 < 8x - 4\)[/tex]
Step 1: Add 4 to both sides to isolate the term with [tex]\( x \)[/tex].
[tex]\[ -68 + 4 < 8x - 4 + 4 \][/tex]
[tex]\[ -64 < 8x \][/tex]
Step 2: Divide both sides by 8.
[tex]\[ \frac{-64}{8} < x \][/tex]
[tex]\[ -8 < x \][/tex]
So, the first part of the inequality simplifies to:
[tex]\[ x > -8 \][/tex]
#### Part B: [tex]\(8x - 4 < -36\)[/tex]
Step 1: Add 4 to both sides to isolate the term with [tex]\( x \)[/tex].
[tex]\[ 8x - 4 + 4 < -36 + 4 \][/tex]
[tex]\[ 8x < -32 \][/tex]
Step 2: Divide both sides by 8.
[tex]\[ \frac{8x}{8} < \frac{-32}{8} \][/tex]
[tex]\[ x < -4 \][/tex]
### 2. Combine the results:
Now we combine the results from Part A and Part B.
Given:
[tex]\[ x > -8 \][/tex]
[tex]\[ x < -4 \][/tex]
We combine these into a single compound inequality:
[tex]\[ -8 < x < -4 \][/tex]
So, the solution to the inequality [tex]\(-68 < 8x - 4 < -36\)[/tex] is:
[tex]\[ \boxed{-8 < x < -4} \][/tex]
