To solve the compound inequality [tex]\(-13 \leq 3 + 8p \leq 11\)[/tex], we need to break it down into two separate inequalities and solve each one step by step.
### Step 1: Solve the first part of the inequality [tex]\(-13 \leq 3 + 8p\)[/tex]
1. Subtract 3 from both sides to isolate the term with [tex]\(p\)[/tex]:
[tex]\[
-13 - 3 \leq 8p
\][/tex]
Simplifying this, we get:
[tex]\[
-16 \leq 8p
\][/tex]
2. Divide both sides by 8 to solve for [tex]\(p\)[/tex]:
[tex]\[
\frac{-16}{8} \leq p
\][/tex]
Simplifying this, we get:
[tex]\[
-2 \leq p
\][/tex]
### Step 2: Solve the second part of the inequality [tex]\(3 + 8p \leq 11\)[/tex]
1. Subtract 3 from both sides to isolate the term with [tex]\(p\)[/tex]:
[tex]\[
3 + 8p - 3 \leq 11 - 3
\][/tex]
Simplifying this, we get:
[tex]\[
8p \leq 8
\][/tex]
2. Divide both sides by 8 to solve for [tex]\(p\)[/tex]:
[tex]\[
\frac{8}{8} \leq p
\][/tex]
Simplifying this, we get:
[tex]\[
p \leq 1
\][/tex]
### Step 3: Combine the solutions
From the first part, we have [tex]\( -2 \leq p \)[/tex]. From the second part, we have [tex]\( p \leq 1 \)[/tex]. Combining these two results gives us the final solution:
[tex]\[
-2 \leq p \leq 1
\][/tex]
Therefore, the solution to the compound inequality [tex]\(-13 \leq 3 + 8p \leq 11\)[/tex] is:
[tex]\[
p \in [-2, 1]
\][/tex]
