To solve the compound inequality [tex]\(3x - 8 \geq -5\)[/tex] and [tex]\(2x - 7 < 5\)[/tex], we need to solve each inequality separately and then combine their solutions.
### Step 1: Solve [tex]\(3x - 8 \geq -5\)[/tex]
1. Add 8 to both sides of the inequality:
[tex]\[
3x - 8 + 8 \geq -5 + 8
\][/tex]
Simplifying, we get:
[tex]\[
3x \geq 3
\][/tex]
2. Divide both sides by 3:
[tex]\[
\frac{3x}{3} \geq \frac{3}{3}
\][/tex]
Simplifying, we get:
[tex]\[
x \geq 1
\][/tex]
### Step 2: Solve [tex]\(2x - 7 < 5\)[/tex]
1. Add 7 to both sides of the inequality:
[tex]\[
2x - 7 + 7 < 5 + 7
\][/tex]
Simplifying, we get:
[tex]\[
2x < 12
\][/tex]
2. Divide both sides by 2:
[tex]\[
\frac{2x}{2} < \frac{12}{2}
\][/tex]
Simplifying, we get:
[tex]\[
x < 6
\][/tex]
### Step 3: Combine the Solutions
We need the values of [tex]\(x\)[/tex] that satisfy both inequalities:
- From the first inequality, [tex]\(3x - 8 \geq -5\)[/tex], we have [tex]\(x \geq 1\)[/tex].
- From the second inequality, [tex]\(2x - 7 < 5\)[/tex], we have [tex]\(x < 6\)[/tex].
The intersection of [tex]\(x \geq 1\)[/tex] and [tex]\(x < 6\)[/tex] is:
[tex]\[
1 \leq x < 6
\][/tex]
Therefore, the final solution to the compound inequality is:
[tex]\[
1 \leq x < 6
\][/tex]
So, the correct answer is:
[tex]\( \boxed{1 \leq x < 6} \)[/tex]
