Sure! Let's solve the inequality step-by-step and then graph the solution set on the number line.
### Step-by-Step Solution
1. Start with the given inequality:
[tex]\[
-4(x + 3) \leq -2x
\][/tex]
2. Distribute [tex]\(-4\)[/tex] on the left side:
[tex]\[
-4x - 12 \leq -2x
\][/tex]
3. To isolate [tex]\(x\)[/tex], add [tex]\(4x\)[/tex] to both sides:
[tex]\[
-4x - 12 + 4x \leq -2x + 4x
\][/tex]
Simplifying, we get:
[tex]\[
-12 \leq 2x
\][/tex]
4. Divide both sides by [tex]\(2\)[/tex]:
[tex]\[
\frac{-12}{2} \leq x
\][/tex]
Simplifying, we get:
[tex]\[
-6 \leq x
\][/tex]
5. Express the solution in interval notation:
[tex]\[
x \geq -6
\][/tex]
Therefore, the solution set to the inequality [tex]\(-4(x + 3) \leq -2x\)[/tex] is:
[tex]\[
[-6, \infty)
\][/tex]
### Graph the Solution Set on the Number Line
1. Draw a number line.
2. Mark the point [tex]\(-6\)[/tex] on the number line.
3. Since the inequality is [tex]\(\geq -6\)[/tex], place a closed circle at [tex]\(-6\)[/tex] to indicate that [tex]\(-6\)[/tex] is included in the solution set.
4. Shade the number line to the right of [tex]\(-6\)[/tex] to indicate all numbers greater than or equal to [tex]\(-6\)[/tex].
### Number Line Representation
```
-∞---|---|---|---|---|---|---|---|---|---|---|---∞
-10 -9 -8 -7 [tex]\(-6\)[/tex] -5 -4 -3 -2 -1 0 ...
Closed circle at -6, and shading to the right:
*========>
```
This graph represents all [tex]\(x\)[/tex] values from [tex]\(-6\)[/tex] to [tex]\(\infty\)[/tex].
