Sure, let's solve the compound inequality step by step.
We're given the inequality:
[tex]\[ -7 < 3x + 2 \leq 14 \][/tex]
We'll break it into two separate inequalities and solve each one.
### Step 1: Solve the inequality [tex]\(-7 < 3x + 2\)[/tex]:
1. Subtract 2 from both sides of the inequality:
[tex]\[
-7 - 2 < 3x + 2 - 2
\][/tex]
Simplifies to:
[tex]\[
-9 < 3x
\][/tex]
2. Divide both sides by 3:
[tex]\[
\frac{-9}{3} < x
\][/tex]
Simplifies to:
[tex]\[
-3 < x
\][/tex]
### Step 2: Solve the inequality [tex]\(3x + 2 \leq 14\)[/tex]:
1. Subtract 2 from both sides of the inequality:
[tex]\[
3x + 2 - 2 \leq 14 - 2
\][/tex]
Simplifies to:
[tex]\[
3x \leq 12
\][/tex]
2. Divide both sides by 3:
[tex]\[
\frac{3x}{3} \leq \frac{12}{3}
\][/tex]
Simplifies to:
[tex]\[
x \leq 4
\][/tex]
### Step 3: Combine the two inequalities:
We now combine the results from step 1 and step 2:
[tex]\[ -3 < x \leq 4 \][/tex]
This means that [tex]\(x\)[/tex] lies within the interval [tex]\((-3, 4]\)[/tex].
So, the solution to the inequality [tex]\(-7 < 3x + 2 \leq 14\)[/tex] is:
[tex]\[ -3 < x \leq 4 \][/tex]
