To solve the compound inequality [tex]\(8 > 3x - 1 > 5\)[/tex], we will break it down into two separate inequalities and solve each one step by step.
### Step 1: Break Down the Inequality
The given compound inequality can be separated into two individual inequalities:
1. [tex]\(8 > 3x - 1\)[/tex]
2. [tex]\(3x - 1 > 5\)[/tex]
### Step 2: Solve the First Inequality
Inequality 1: [tex]\(8 > 3x - 1\)[/tex]
1. Add 1 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
8 + 1 > 3x - 1 + 1
\][/tex]
2. Simplify:
[tex]\[
9 > 3x
\][/tex]
3. Divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{9}{3} > x
\][/tex]
4. Simplify:
[tex]\[
3 > x \quad \text{or} \quad x < 3
\][/tex]
### Step 3: Solve the Second Inequality
Inequality 2: [tex]\(3x - 1 > 5\)[/tex]
1. Add 1 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
3x - 1 + 1 > 5 + 1
\][/tex]
2. Simplify:
[tex]\[
3x > 6
\][/tex]
3. Divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x > \frac{6}{3}
\][/tex]
4. Simplify:
[tex]\[
x > 2
\][/tex]
### Step 4: Combine the Results
We have found two inequalities:
- [tex]\(x < 3\)[/tex]
- [tex]\(x > 2\)[/tex]
Combining these two gives:
[tex]\[
2 < x < 3
\][/tex]
### Final Answer
The correct option is:
b. [tex]\(3 > x > 2\)[/tex] or equivalently [tex]\(2 < x < 3\)[/tex]
