Question

solve the inequality 6(x - 2) > -8(x + 3)
which graph represents this inequality?
a graph
b graph
c graph
d graph

Explanation:

Step1: Solve the inequality \( 4(x - 2) < -8(x + 3) \)

First, distribute the numbers outside the parentheses:
\( 4x - 8 < -8x - 24 \)

Step2: Add \( 8x \) to both sides

To get all the \( x \)-terms on one side, we add \( 8x \) to both sides:
\( 4x + 8x - 8 < -24 \)
\( 12x - 8 < -24 \)

Step3: Add 8 to both sides

Now, add 8 to both sides to isolate the term with \( x \):
\( 12x < -24 + 8 \)
\( 12x < -16 \)

Step4: Divide by 12

Divide both sides by 12 to solve for \( x \):
\( x < \frac{-16}{12} \)
Simplify the fraction: \( x < -\frac{4}{3} \) (or \( x < -1.\overline{3} \))

Now, we need to determine which graph represents \( x < -\frac{4}{3} \). The graph should have an open circle at \( -\frac{4}{3} \) (since the inequality is strict, \( < \), not \( \leq \)) and the arrow pointing to the left (for values less than \( -\frac{4}{3} \)).

Assuming the options are labeled with graphs:

• Option A: Likely has an open circle at a value greater than \( -\frac{4}{3} \) or arrow direction wrong.
• Option B: Maybe incorrect.
• Option C: If it has an open circle at \( -\frac{4}{3} \) and arrow left, but wait, let's re - check. Wait, \( -\frac{4}{3}\approx - 1.333\).
• Option D: Wait, maybe I made a mistake in the initial inequality. Wait, the original inequality is \( 4(x - 2)<-8(x + 3)\). Let's re - solve:

\( 4x-8 < - 8x - 24\)

\(4x + 8x< - 24 + 8\)

\(12x < - 16\)

\(x<-\frac{16}{12}=-\frac{4}{3}\approx - 1.333\)

Now, looking at the options (assuming the graphs are number lines):

If option D has an open circle at \( -\frac{4}{3}\) (or around - 1.33) and the arrow pointing to the left (for values less than \( -\frac{4}{3}\)), then that's the correct graph. But since the user's image shows options A, B, C, D with colored boxes, and the inequality solution is \( x < -\frac{4}{3}\), the correct graph should be the one with an open circle at \( -\frac{4}{3}\) and arrow left.

Answer:

The graph that represents \( x < -\frac{4}{3}\) (with an open circle at \( -\frac{4}{3}\) and arrow to the left) is the correct one. If we assume the options are as per the standard number - line graphs for this inequality, the answer is the option (e.g., if D is the one with open circle at \( -\frac{4}{3}\) and left arrow, then D. But since the exact graphs are not fully visible, based on the solution of the inequality \( x < -\frac{4}{3}\), the correct graph is the one with open circle at \( -\frac{4}{3}\) and arrow pointing left.