To solve the given inequality, we need to consider both parts of the inequality separately and then combine the solutions.
### Part 1: Solving [tex]\( 11x + 22 \leq -44 \)[/tex]
1. Subtract 22 from both sides:
[tex]\[
11x + 22 - 22 \leq -44 - 22 \implies 11x \leq -66
\][/tex]
2. Divide both sides by 11:
[tex]\[
x \leq -6
\][/tex]
### Part 2: Solving [tex]\( 8x + 14 > 30 \)[/tex]
1. Subtract 14 from both sides:
[tex]\[
8x + 14 - 14 > 30 - 14 \implies 8x > 16
\][/tex]
2. Divide both sides by 8:
[tex]\[
x > 2
\][/tex]
### Combining the Solutions
From part 1, we have [tex]\( x \leq -6 \)[/tex].
From part 2, we have [tex]\( x > 2 \)[/tex].
Since these are independent inequalities connected by an "or" condition, we combine both solutions to form the union of the intervals:
[tex]\[
(-\infty, -6] \cup (2, \infty)
\][/tex]
### Conclusion
The solution to the inequality [tex]\( 11x + 22 \leq -44 \)[/tex] or [tex]\( 8x + 14 > 30 \)[/tex] is given by the interval:
[tex]\[
(-\infty, -6] \cup (2, \infty)
\][/tex]
Therefore, the correct choice is:
b. [tex]\( (-\infty, -6] \cup (2, \infty) \)[/tex]
