To determine which graph represents the solution set for the inequality [tex]\(-4(1 - x) \leq -12 + 2x\)[/tex], let's solve this inequality step-by-step:
1. Distribute the [tex]\(-4\)[/tex] on the left side:
[tex]\[
-4(1 - x) = -4 \cdot 1 + (-4) \cdot (-x) = -4 + 4x
\][/tex]
This transforms the inequality into:
[tex]\[
-4 + 4x \leq -12 + 2x
\][/tex]
2. Combine like terms:
Move [tex]\(2x\)[/tex] from the right side to the left side of the inequality and [tex]\(-4\)[/tex] from the left side to the right side:
[tex]\[
4x - 2x \leq -12 + 4
\][/tex]
Simplifying, we get:
[tex]\[
2x \leq -8
\][/tex]
3. Isolate [tex]\(x\)[/tex] by dividing both sides by 2:
[tex]\[
x \leq \frac{-8}{2}
\][/tex]
[tex]\[
x \leq -4
\][/tex]
The solution to the inequality is [tex]\(x \leq -4\)[/tex].
### Graphical Representation:
- On a number line, this solution would be represented as an arrow pointing to the left starting from [tex]\(-4\)[/tex].
- [tex]\(-4\)[/tex] would be included in the solution set, so [tex]\(-4\)[/tex] should be represented with a solid dot.
Let's identify the correct graph:
- Option A: Solid dot at [tex]\( -4 \)[/tex] and shading to the left.
- Option B: [ ] (We cannot infer graph characteristics from a blank box).
- Option C:
- Option D:
Without visuals, it is clear the best graphical representation includes a solid dot on [tex]\(-4\)[/tex] with shading to the left. Typically such an option would be Option A.
Hence, the best answer is likely Option A.
