To solve the compound inequality [tex]\(x - 2 < -11\)[/tex] or [tex]\(-3x \leq -18\)[/tex], let's solve each part step-by-step:
### Part 1: [tex]\(x - 2 < -11\)[/tex]
1. Start with the inequality:
[tex]\[ x - 2 < -11 \][/tex]
2. Add 2 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x - 2 + 2 < -11 + 2 \][/tex]
[tex]\[ x < -9 \][/tex]
### Part 2: [tex]\(-3x \leq -18\)[/tex]
1. Start with the inequality:
[tex]\[ -3x \leq -18 \][/tex]
2. Divide both sides by -3. Remember to reverse the inequality symbol when dividing by a negative number:
[tex]\[ x \geq \frac{-18}{-3} \][/tex]
[tex]\[ x \geq 6 \][/tex]
### Combining the Inequalities
Now that we have solved each individual inequality, we combine the results. The solution to the original compound inequality [tex]\(x - 2 < -11\)[/tex] or [tex]\(-3x \leq -18\)[/tex] is:
[tex]\[ x < -9 \quad \text{or} \quad x \geq 6 \][/tex]
So, the final solution is:
[tex]\[ x < -9 \quad \text{or} \quad x \geq 6 \][/tex]
This means that [tex]\(x\)[/tex] can be any number less than -9 or any number greater than or equal to 6.
