Let's solve the compound inequality step by step.
We have two inequalities:
1. [tex]\(2x + 6 > -11\)[/tex]
2. [tex]\(2x + 6 < 10\)[/tex]
We need to solve each inequality separately.
### Solving the first inequality:
[tex]\[2x + 6 > -11\][/tex]
Step 1: Subtract 6 from both sides to isolate the term with the variable [tex]\(x\)[/tex]:
[tex]\[
2x + 6 - 6 > -11 - 6
\][/tex]
[tex]\[
2x > -17
\][/tex]
Step 2: Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{2x}{2} > \frac{-17}{2}
\][/tex]
[tex]\[
x > -\frac{17}{2}
\][/tex]
### Solving the second inequality:
[tex]\[2x + 6 < 10\][/tex]
Step 1: Subtract 6 from both sides to isolate the term with the variable [tex]\(x\)[/tex]:
[tex]\[
2x + 6 - 6 < 10 - 6
\][/tex]
[tex]\[
2x < 4
\][/tex]
Step 2: Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{2x}{2} < \frac{4}{2}
\][/tex]
[tex]\[
x < 2
\][/tex]
### Combining the results:
We now combine the solutions from both inequalities:
[tex]\[
-\frac{17}{2} < x < 2
\][/tex]
This means [tex]\(x\)[/tex] must be greater than [tex]\(-\frac{17}{2}\)[/tex] and less than 2. The solution set can be written in interval notation as:
[tex]\[
\left( -\frac{17}{2}, 2 \right)
\][/tex]
### Conclusion:
The solution set for the compound inequality [tex]\(2x + 6 > -11\)[/tex] and [tex]\(2x + 6 < 10\)[/tex] is [tex]\(\left( -\frac{17}{2}, 2 \right)\)[/tex].
