To determine which graph represents the solution to the inequality [tex]\(-\frac{1}{4}(12x + 8) \leq -2x + 11\)[/tex], we first need to solve it. Here's the process broken down step-by-step:
1. Distribute and Simplify the Inequality:
Simplifying the left-hand side of the inequality:
[tex]\[
-\frac{1}{4}(12x + 8) \leq -2x + 11
\][/tex]
Distribute [tex]\(-\frac{1}{4}\)[/tex] over [tex]\(12x + 8\)[/tex]:
[tex]\[
-\frac{1}{4} \cdot 12x - \frac{1}{4} \cdot 8 \leq -2x + 11
\][/tex]
This simplifies to:
[tex]\[
-3x - 2 \leq -2x + 11
\][/tex]
2. Rearrange and Combine Like Terms:
Add [tex]\(2x\)[/tex] to both sides to start solving for [tex]\(x\)[/tex]:
[tex]\[
-3x + 2x - 2 \leq 11
\][/tex]
Simplifying gives:
[tex]\[
-x - 2 \leq 11
\][/tex]
3. Continue Isolating [tex]\(x\)[/tex]:
Add 2 to both sides to fully isolate [tex]\(x\)[/tex]:
[tex]\[
-x - 2 + 2 \leq 11 + 2
\][/tex]
This simplifies to:
[tex]\[
-x \leq 13
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Multiply both sides by [tex]\(-1\)[/tex] (remember to reverse the inequality sign):
[tex]\[
x \geq -13
\][/tex]
This inequality can be written as:
[tex]\[ -13 \leq x \][/tex]
5. Determine the Representation:
The solution [tex]\( -13 \leq x \)[/tex] indicates that [tex]\(x\)[/tex] is any number greater than or equal to [tex]\(-13\)[/tex].
6. Interpret on the Graph:
This means on a number line or coordinate plane, you’ll have a solid dot (or inclusive bracket) at [tex]\(-13\)[/tex] and shading or a line extending to the right towards positive infinity.
Given the above steps, the correct graphical representation should show:
- An inclusive point or a solid dot at [tex]\( x = -13 \)[/tex]
- A shading or line extending to the right from [tex]\( x = -13 \)[/tex].
Among the given options:
- A graph starting from [tex]\(-13\)[/tex] and extending to the right (including [tex]\(-13\)[/tex] itself).
Ensure the graph correctly follows these specifications to select the appropriate option. The correct answer is likely one that represents [tex]\( -13 \leq x < \infty \)[/tex].
