Certainly! Let's solve the compound inequality step-by-step.
The given compound inequality is:
[tex]\[ 8 > 3x - 1 > 5 \][/tex]
This compound inequality consists of two separate inequalities:
1. [tex]\( 8 > 3x - 1 \)[/tex]
2. [tex]\( 3x - 1 > 5 \)[/tex]
We'll solve each inequality individually and then find the intersection of the solutions.
### Solving the first inequality:
[tex]\[ 8 > 3x - 1 \][/tex]
Add 1 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 8 + 1 > 3x \][/tex]
[tex]\[ 9 > 3x \][/tex]
Divide both sides by 3:
[tex]\[ \frac{9}{3} > x \][/tex]
[tex]\[ 3 > x \][/tex]
This can be rewritten as:
[tex]\[ x < 3 \][/tex]
### Solving the second inequality:
[tex]\[ 3x - 1 > 5 \][/tex]
Add 1 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 3x - 1 + 1 > 5 + 1 \][/tex]
[tex]\[ 3x > 6 \][/tex]
Divide both sides by 3:
[tex]\[ \frac{3x}{3} > \frac{6}{3} \][/tex]
[tex]\[ x > 2 \][/tex]
### Combining both solutions:
From the first inequality, we have:
[tex]\[ x < 3 \][/tex]
From the second inequality, we have:
[tex]\[ x > 2 \][/tex]
Combining these, we get:
[tex]\[ 2 < x < 3 \][/tex]
In interval notation, this is [tex]\( (2, 3) \)[/tex].
Now, let's match this to the given choices:
a. [tex]\( \frac{7}{3} > x > \frac{4}{3} \)[/tex]
b. [tex]\( 12 < x < 28 \)[/tex]
c. [tex]\( 3 > x > 2 \)[/tex]
d. [tex]\( 27 > x > 18 \)[/tex]
Clearly, the correct choice is:
c. [tex]\( 3 > x > 2 \)[/tex]
Thus, the solution to the compound inequality [tex]\( 8 > 3x - 1 > 5 \)[/tex] is:
[tex]\[ 3 > x > 2 \][/tex]
So, the correct answer is:
c. [tex]\( 3 > x > 2 \)[/tex]
