Question

the solution to which inequality is graphed below?
options:
6b < -18
3b > -1
-6b > 18
4b > -12

Explanation:

Step1: Analyze the number line

The number line has an open circle at -3 (wait, no, looking at the numbers, the marked point seems to be at -3? Wait, the numbers on the line: let's check the inequality solutions. Wait, first, let's solve each inequality:

1. For \(6b < -18\): Divide both sides by 6: \(b < -3\)
2. For \(3b > -1\): Divide by 3: \(b > -\frac{1}{3}\)
3. For \(-6b > 18\): Divide by -6 (reverse inequality): \(b < -3\) (Wait, no: \(-6b > 18\) → divide by -6, so \(b < -3\) (since dividing by negative flips inequality)
4. For \(4b > -12\): Divide by 4: \(b > -3\)

Wait, the number line: the blue line is to the left of -3 (open circle? Wait, the original graph: the marked point is at -3, and the line is extending to the left (since the arrow is towards lower numbers). Wait, let's re-express:

Wait, the options:

First, solve each inequality:

• \(6b < -18\): \(b < -3\) (divide by 6)
• \(3b > -1\): \(b > -1/3\)
• \(-6b > 18\): divide by -6, so \(b < -3\) (because -6b > 18 → b < 18/(-6) → b < -3)
• \(4b > -12\): \(b > -3\)

Now, the number line: the graph has an open circle at -3 (assuming) and the line is to the left (so \(b < -3\)). Now check which inequality gives \(b < -3\):

• \(6b < -18\): \(b < -3\) (correct)
• \(-6b > 18\): \(b < -3\) (also correct? Wait, let's check the signs. Wait, \(-6b > 18\): multiply both sides by -1: \(6b < -18\), which is the same as the first inequality. Wait, maybe the original graph: let's check the numbers. Wait, the user's image: the number line has marks, and the solution is to the left of -3 (open circle). So let's check each option:

Option 1: \(6b < -18\) → \(b < -3\) (matches left of -3)

Option 3: \(-6b > 18\) → \(b < -3\) (also matches). Wait, but maybe I misread the number line. Wait, the original problem's number line: let's see the numbers. The first option is \(6b < -18\), solving gives \(b < -3\). Let's confirm.

Wait, the number line: the blue line is from -3 (open circle) to the left (towards more negative numbers). So the solution is \(b < -3\). Now, solve each inequality:

• \(6b < -18\): \(b < -3\) (correct)
• \(-6b > 18\): \(b < -3\) (also correct? Wait, let's compute \(-6b > 18\): if b = -4, then -6(-4)=24 > 18: yes. If b = -3, -6(-3)=18, which is not >18, so open circle. So \(-6b > 18\) also gives \(b < -3\). Wait, but the first option is \(6b < -18\): if b = -4, 6(-4)=-24 < -18: yes. If b = -3, 6(-3)=-18, which is not < -18, so open circle. So both \(6b < -18\) and \(-6b > 18\) give \(b < -3\). Wait, but maybe the original graph's numbering: let's check the options again. Wait, the first option is \(6b < -18\), which is a common inequality. Let's proceed.

Wait, maybe I made a mistake with \(-6b > 18\): let's re-express. \(-6b > 18\) is equivalent to \(6b < -18\) (multiply both sides by -1, reverse inequality). So both are same. But let's check the options. The first option is \(6b < -18\), which solves to \(b < -3\), matching the number line (line to the left of -3).

Step2: Confirm the solution

The number line shows \(b < -3\). Solving \(6b < -18\) gives \(b < -3\) (divide both sides by 6, no inequality flip). So this matches the graph.

Answer:

\(6b < -18\) (the first option, assuming the options are A. \(6b < -18\), B. \(3b > -1\), C. \(-6b > 18\), D. \(4b > -12\); but based on the calculation, \(6b < -18\) gives \(b < -3\) which matches the number line graph.)