Let's solve the compound inequality step by step:
### Inequality 1: [tex]\(5x - 7 \geq 8\)[/tex]
1. Isolate the variable term:
Add 7 to both sides of the inequality:
[tex]\[
5x - 7 + 7 \geq 8 + 7
\][/tex]
Simplifying this, we get:
[tex]\[
5x \geq 15
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
Divide both sides by 5:
[tex]\[
\frac{5x}{5} \geq \frac{15}{5}
\][/tex]
Simplifying this, we get:
[tex]\[
x \geq 3
\][/tex]
### Inequality 2: [tex]\(4x + 4 < -20\)[/tex]
1. Isolate the variable term:
Subtract 4 from both sides of the inequality:
[tex]\[
4x + 4 - 4 < -20 - 4
\][/tex]
Simplifying this, we get:
[tex]\[
4x < -24
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
Divide both sides by 4:
[tex]\[
\frac{4x}{4} < \frac{-24}{4}
\][/tex]
Simplifying this, we get:
[tex]\[
x < -6
\][/tex]
### Combine the results
From the two inequalities, we have:
1. [tex]\( x \geq 3 \)[/tex]
2. [tex]\( x < -6 \)[/tex]
Thus, the solution to the compound inequality [tex]\( 5x - 7 \geq 8 \)[/tex] or [tex]\( 4x + 4 < -20 \)[/tex] is:
[tex]\[ x \geq 3 \text{ or } x < -6 \][/tex]
Therefore, the correct choice is:
b. [tex]\( x \geq 3 \text{ or } x < -6 \)[/tex]
