To solve the compound inequality [tex]\( -11 < 2x + 5 < 17 \)[/tex], we need to break it down into two separate inequalities and solve each one step-by-step.
1. Solve the first inequality:
[tex]\[
-11 < 2x + 5
\][/tex]
- Begin by isolating [tex]\( 2x \)[/tex] by subtracting 5 from both sides:
[tex]\[
-11 - 5 < 2x
\][/tex]
Simplifying the left side:
[tex]\[
-16 < 2x
\][/tex]
- Next, divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{-16}{2} < x
\][/tex]
Simplifying:
[tex]\[
-8 < x
\][/tex]
2. Solve the second inequality:
[tex]\[
2x + 5 < 17
\][/tex]
- Subtract 5 from both sides:
[tex]\[
2x < 17 - 5
\][/tex]
Simplifying the right side:
[tex]\[
2x < 12
\][/tex]
- Next, divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{12}{2} > x
\][/tex]
Simplifying:
[tex]\[
x < 6
\][/tex]
3. Combine the two inequalities:
- From step 1, we have [tex]\( -8 < x \)[/tex]
- From step 2, we have [tex]\( x < 6 \)[/tex]
Combining these results, we get the compound inequality:
[tex]\[
-8 < x < 6
\][/tex]
So, the correct answer is:
b. [tex]\( -8 < x < 6 \)[/tex]
