Let's solve the inequality step-by-step and graph the solution on a number line.
Step 1: Start with the given inequality:
[tex]\[
-4(x + 3) \leq -2x
\][/tex]
Step 2: Distribute the [tex]\(-4\)[/tex] on the left side:
[tex]\[
-4x - 12 \leq -2x
\][/tex]
Step 3: Add [tex]\(4x\)[/tex] to both sides to combine like terms:
[tex]\[
-4x + 4x - 12 \leq -2x + 4x
\][/tex]
This simplifies to:
[tex]\[
-12 \leq 2x
\][/tex]
Step 4: Divide both sides by [tex]\(2\)[/tex]:
[tex]\[
\frac{-12}{2} \leq \frac{2x}{2}
\][/tex]
This simplifies to:
[tex]\[
-6 \leq x
\][/tex]
Step 5: Rewrite the inequality in a more conventional form:
[tex]\[
x \geq -6
\][/tex]
Now, let's graph this solution on a number line:
1. Draw a number line with appropriate markings.
2. Locate [tex]\(-6\)[/tex] on the number line.
3. Since the inequality is [tex]\(x \geq -6\)[/tex], we use a closed circle at [tex]\(-6\)[/tex] to indicate that [tex]\(-6\)[/tex] is included in the solution set.
4. Shade the region to the right of [tex]\(-6\)[/tex] to indicate all values greater than or equal to [tex]\(-6\)[/tex].
Here is what the number line looks like:
[tex]\[
\begin{array}{c}
\longleftarrow \quad \; \; \; \; \; \; \; \; \; \; \; \; \circled{-6} \blacksquare \rightarrow \quad \; \; \; \; \; \; \longrightarrow\\
\end{array}
\][/tex]
The closed circle ([tex]\(\blacksquare\)[/tex]) at [tex]\(-6\)[/tex] indicates that [tex]\(-6\)[/tex] is included in the solution set, and the arrow to the right shows that all values greater than or equal to [tex]\(-6\)[/tex] are part of the solution.
