To solve the compound inequality [tex]\(-7 \leq 6x + 5 \leq 23\)[/tex], we will follow these steps:
1. Isolate the term involving [tex]\(x\)[/tex]:
We start with the inequality:
[tex]\[
-7 \leq 6x + 5 \leq 23
\][/tex]
To isolate [tex]\(6x\)[/tex], we need to eliminate the constant term [tex]\(+5\)[/tex] that is added to [tex]\(6x\)[/tex]. We do this by subtracting 5 from all three parts of the inequality:
[tex]\[
-7 - 5 \leq 6x + 5 - 5 \leq 23 - 5
\][/tex]
Simplifying each part gives us:
[tex]\[
-12 \leq 6x \leq 18
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
Now that we have [tex]\(-12 \leq 6x \leq 18\)[/tex], we need to solve for [tex]\(x\)[/tex] by undoing the multiplication by 6. We do this by dividing all three parts of the inequality by 6:
[tex]\[
\frac{-12}{6} \leq \frac{6x}{6} \leq \frac{18}{6}
\][/tex]
Simplifying each part gives us:
[tex]\[
-2 \leq x \leq 3
\][/tex]
So, the solution to the compound inequality [tex]\(-7 \leq 6x + 5 \leq 23\)[/tex] is:
[tex]\[
-2 \leq x \leq 3
\][/tex]
Therefore, the correct answer is:
d. [tex]\(-2 \leq x \leq 3\)[/tex]
