Sure, let's solve the inequality step by step to identify the correct graph representation.
Firstly, let's distribute the [tex]\(-\frac{1}{4}\)[/tex] on the left side of the inequality:
[tex]\[ -\frac{1}{4}(12x + 8) \leq -2x + 11 \][/tex]
Distributing [tex]\(-\frac{1}{4}\)[/tex], we get:
[tex]\[ -3x - 2 \leq -2x + 11 \][/tex]
Next, we need to isolate the variable [tex]\(x\)[/tex] on one side of the inequality. So, let's add [tex]\(3x\)[/tex] to both sides:
[tex]\[ -2 \leq -2x + 3x + 11 \][/tex]
This simplifies to:
[tex]\[ -2 \leq x + 11 \][/tex]
Now, subtract 11 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[ -2 - 11 \leq x \][/tex]
Simplifying further, we get:
[tex]\[ -13 \leq x \][/tex]
or equivalently,
[tex]\[ x \geq -13 \][/tex]
So the solution to the inequality is [tex]\(x \geq -13\)[/tex].
Now, let's determine which graph represents this solution. We are looking for a graph with a number line where the shading starts at [tex]\(-13\)[/tex] and extends to the right, indicating that [tex]\(x\)[/tex] includes [tex]\(-13\)[/tex] and all numbers greater than [tex]\(-13\)[/tex].
A. If the graph shows a closed circle at [tex]\(-13\)[/tex] and shading to the right, including all numbers greater than or equal to [tex]\(-13\)[/tex], then this is correct.
B. If the graph does not match this, it is incorrect.
C. If the graph does not match this, it is incorrect.
D. If the graph matches this, it is correct.
Based on the correct description, the answer is the graph with a closed circle at [tex]\(-13\)[/tex] and shading to the right, indicating [tex]\(x \geq -13\)[/tex]. Select the graph accordingly.
