Question

which number line shows the solution to: 11x + 14 < -8? (with four number lines labeled a, b, c, d and options a, b, c, d)

Explanation:

Step1: Solve the inequality for x

Start with the inequality \(11x + 14 < -8\). Subtract 14 from both sides:
\(11x + 14 - 14 < -8 - 14\)
Simplify to get \(11x < -22\). Then divide both sides by 11:
\(\frac{11x}{11} < \frac{-22}{11}\)
Which simplifies to \(x < -2\).

Step2: Analyze the number lines

We need a number line with an open circle at -2 (since the inequality is strict, \(x < -2\), not \(x \leq -2\)) and the line shaded to the left of -2 (because x is less than -2). Looking at the options, we check the position of the open circle and the direction of shading. The number line that has an open circle at -2 and shading to the left (towards more negative numbers) is the correct one. From the given options (assuming the labels: let's check the number lines. If we consider the options, the one with open circle at -2 and shading left is the correct representation for \(x < -2\)).

Answer:

The number line (let's assume the correct one is the one with open circle at -2 and shading to the left, corresponding to the option that matches \(x < -2\); based on the process, the correct number line should have open circle at -2 and left shading, so among the options, the one that fits is the one with open circle at -2 and line to the left, e.g., if D (or the relevant option) has open circle at -2 and shading left, then the answer is that option. But from the solving, \(x < -2\), so the number line with open circle at -2 and shaded left is correct. So the answer is the number line (e.g., if D is that, then D. [description of D's number line with open circle at -2 and left shading]).

(Note: Since the exact labels are a bit unclear from the image, but the key is the solution \(x < -2\) leads to open circle at -2, left shading. So the correct number line is the one with open circle at -2 and shading to the left.)