Solve the inequality.

[tex]\[
-\frac{1}{3} x - 12 \ \textgreater \ 21 \quad \text{or} \quad -6x + 10 \ \textless \ -2
\][/tex]

[tex]\[
x \ \textless \ [?] \quad \text{or} \quad x \ \textgreater \ \square
\][/tex]



Answer :

Certainly! Let's solve the given inequalities step-by-step.

### First Inequality: [tex]\(-\frac{1}{3}x - 12 > 21\)[/tex]

1. Isolate the term involving [tex]\(x\)[/tex]:
[tex]\[ -\frac{1}{3}x - 12 > 21 \][/tex]
Add 12 to both sides:
[tex]\[ -\frac{1}{3}x > 33 \][/tex]

2. Solve for [tex]\(x\)[/tex]:
Multiply both sides by [tex]\(-3\)[/tex], and remember to flip the inequality sign when multiplying by a negative number:
[tex]\[ x < -99 \][/tex]

So, the first inequality solution is:
[tex]\[ x < -99 \][/tex]

### Second Inequality: [tex]\(-6x + 10 < -2\)[/tex]

1. Isolate the term involving [tex]\(x\)[/tex]:
[tex]\[ -6x + 10 < -2 \][/tex]
Subtract 10 from both sides:
[tex]\[ -6x < -12 \][/tex]

2. Solve for [tex]\(x\)[/tex]:
Divide both sides by [tex]\(-6\)[/tex], and remember to flip the inequality sign when dividing by a negative number:
[tex]\[ x > 2 \][/tex]

So, the second inequality solution is:
[tex]\[ x > 2 \][/tex]

### Combined Solution
The combined solution for the inequalities is:
[tex]\[ x < -99 \quad \text{or} \quad x > 2 \][/tex]