To solve the compound inequality:
[tex]\[ -7 \leq 6x + 5 \leq 23 \][/tex]
we need to address each part of the compound inequality separately.
1. Solve the left inequality:
[tex]\[ -7 \leq 6x + 5 \][/tex]
First, isolate [tex]\(6x\)[/tex] by subtracting 5 from both sides:
[tex]\[ -7 - 5 \leq 6x \][/tex]
[tex]\[ -12 \leq 6x \][/tex]
Next, divide both sides by 6 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{-12}{6} \leq x \][/tex]
[tex]\[ -2 \leq x \][/tex]
2. Solve the right inequality:
[tex]\[ 6x + 5 \leq 23 \][/tex]
First, isolate [tex]\(6x\)[/tex] by subtracting 5 from both sides:
[tex]\[ 6x + 5 - 5 \leq 23 - 5 \][/tex]
[tex]\[ 6x \leq 18 \][/tex]
Next, divide both sides by 6 to solve for [tex]\(x\)[/tex]:
[tex]\[ \frac{18}{6} \leq x \][/tex]
[tex]\[ x \leq 3 \][/tex]
3. Combine the solutions from both inequalities:
[tex]\[ -2 \leq x \leq 3 \][/tex]
So the solution to the compound inequality is:
[tex]\[ -2 \leq x \leq 3 \][/tex]
By evaluating the provided choices, we can see that the correct interval for [tex]\(x\)[/tex] is:
[tex]\[ \text{c. } -2 \leq x \leq 3 \][/tex]
