Sure, let's solve the inequality step-by-step to find the correct solution.
Given inequality:
[tex]\[
-8x - 1 > 3x + 14
\][/tex]
First, we'll move all terms involving [tex]\(x\)[/tex] to one side and the constant terms to the other side.
1. Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[
-8x - 1 - 3x > 14
\][/tex]
This simplifies to:
[tex]\[
-11x - 1 > 14
\][/tex]
2. Next, add 1 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
-11x - 1 + 1 > 14 + 1
\][/tex]
This simplifies to:
[tex]\[
-11x > 15
\][/tex]
3. To solve for [tex]\(x\)[/tex], we need to divide both sides of the inequality by [tex]\(-11\)[/tex]. Remember that dividing or multiplying both sides of an inequality by a negative number reverses the direction of the inequality:
[tex]\[
x < \frac{15}{11}
\][/tex]
So, the solution set for the inequality [tex]\(-8x - 1 > 3x + 14\)[/tex] is:
[tex]\[
x < \frac{15}{11}
\][/tex]
Among the given choices, this corresponds to:
[tex]\[
x < -\frac{15}{11}
\][/tex]
Therefore, the correct answer is:
[tex]\(x < -\frac{15}{11}\)[/tex].
