To solve the given inequalities:
[tex]\[
-\frac{1}{3} x - 12 > 21 \quad \text{or} \quad -6x + 10 < -2
\][/tex]
we will tackle each inequality step by step.
Step 1: Solve [tex]\(-\frac{1}{3} x - 12 > 21\)[/tex]
1.1. Start by isolating [tex]\(x\)[/tex]:
[tex]\[
-\frac{1}{3} x - 12 > 21
\][/tex]
1.2. Add 12 to both sides to simplify:
[tex]\[
-\frac{1}{3} x > 33
\][/tex]
1.3. Multiply both sides by -3 (note that multiplying by a negative number reverses the inequality sign):
[tex]\[
x < -99
\][/tex]
So, the solution for the first inequality is:
[tex]\[
x < -99
\][/tex]
Step 2: Solve [tex]\(-6x + 10 < -2\)[/tex]
2.1. Start by isolating [tex]\(x\)[/tex]:
[tex]\[
-6x + 10 < -2
\][/tex]
2.2. Subtract 10 from both sides to simplify:
[tex]\[
-6x < -12
\][/tex]
2.3. Divide both sides by -6 (again, remember to reverse the inequality sign):
[tex]\[
x > 2
\][/tex]
So, the solution for the second inequality is:
[tex]\[
x > 2
\][/tex]
Combining both solutions, we get:
[tex]\[
x < -99 \quad \text{or} \quad x > 2
\][/tex]
Therefore, the final solution can be written as:
[tex]\[
x < -99 \quad \text{or} \quad x > 2
\][/tex]
