To solve the inequality [tex]\(-3 < 2x + 1 \leq 11 - 3x\)[/tex], we will break it down into two separate parts and solve each part step by step.
### Step 1: Solving [tex]\(-3 < 2x + 1\)[/tex]
1. Subtract 1 from both sides:
[tex]\[
-3 - 1 < 2x
\][/tex]
Simplifying the left-hand side, we get:
[tex]\[
-4 < 2x
\][/tex]
2. Divide both sides by 2:
[tex]\[
\frac{-4}{2} < x
\][/tex]
Simplifying, we obtain:
[tex]\[
-2 < x
\][/tex]
### Step 2: Solving [tex]\(2x + 1 \leq 11 - 3x\)[/tex]
1. Subtract [tex]\(1\)[/tex] from both sides:
[tex]\[
2x + 1 - 1 \leq 11 - 3x - 1
\][/tex]
Simplifying, we get:
[tex]\[
2x \leq 10 - 3x
\][/tex]
2. Add [tex]\(3x\)[/tex] to both sides to gather all [tex]\(x\)[/tex] terms on one side:
[tex]\[
2x + 3x \leq 10
\][/tex]
Simplifying, we have:
[tex]\[
5x \leq 10
\][/tex]
3. Divide both sides by 5:
[tex]\[
x \leq \frac{10}{5}
\][/tex]
Simplifying, we get:
[tex]\[
x \leq 2
\][/tex]
### Combining the Results
Now, we combine the results from both inequalities:
- From the first inequality, we have [tex]\( -2 < x \)[/tex].
- From the second inequality, we have [tex]\( x \leq 2 \)[/tex].
Combining these gives us the final solution:
[tex]\[
-2 < x \leq 2
\][/tex]
### Conclusion
The solution to the inequality [tex]\( -3 < 2x + 1 \leq 11 - 3x \)[/tex] is:
[tex]\[
D: -2 < x \leq 2
\][/tex]
