Let's solve the inequality step-by-step and then represent the solution on the number line.
### Step 1: Simplify the Inequality
First, we need to distribute the -4 on the left-hand side.
[tex]\[
-4(x + 3) \leq -2x
\][/tex]
This simplifies to:
[tex]\[
-4x - 12 \leq -2x
\][/tex]
### Step 2: Isolate the Variable x
Next, we want to get all terms involving [tex]\( x \)[/tex] on one side and constants on the other. We can do this by adding [tex]\( 4x \)[/tex] to both sides:
[tex]\[
-4x - 12 + 4x \leq -2x + 4x
\][/tex]
This simplifies to:
[tex]\[
-12 \leq 2x
\][/tex]
### Step 3: Solve for x
Now, we need to isolate [tex]\( x \)[/tex] by dividing both sides by 2:
[tex]\[
\frac{-12}{2} \leq x
\][/tex]
This simplifies to:
[tex]\[
-6 \leq x
\][/tex]
Or, equivalently:
[tex]\[
x \geq -6
\][/tex]
### Step 4: Graph the Solution on the Number Line
Now we need to represent this solution set on the number line. The inequality [tex]\( x \geq -6 \)[/tex] means that [tex]\( x \)[/tex] is greater than or equal to [tex]\(-6\)[/tex].
1. Draw a number line.
2. Mark the point [tex]\(-6\)[/tex] on the number line.
3. Use a solid dot at [tex]\(-6\)[/tex] to indicate that [tex]\(-6\)[/tex] is included in the solution set (since the inequality is "greater than or equal to").
4. Shade the portion of the number line to the right of [tex]\(-6\)[/tex], extending to infinity, to indicate that all numbers greater than [tex]\(-6\)[/tex] are part of the solution set.
Here is the graphical representation:
[tex]\[
\begin{array}{c}
\begin{tikzpicture}
\draw[->] (-7,0) -- (4,0) node[right] {};
\foreach \x in { -6, -5, -4, -3, -2, -1, 0, 1, 2, 3}
\draw (\x, 0.1) -- (\x, -0.1) node[below] {\x};
\fill[black] (-6,0) circle (2pt);
\draw[thick, -] (-6,0) -- (4,0);
\end{tikzpicture}
\end{array}
\][/tex]
This number line shows the solution set for the inequality [tex]\(-4(x + 3) \leq -2x\)[/tex], which is [tex]\( x \geq -6 \)[/tex].
