Certainly! Let's solve the inequality step-by-step and then graph the solution on a number line.
### Step-By-Step Solution
1. Distribute the -2 on the left side:
[tex]\[
-2(x + 6) > -4x
\][/tex]
This simplifies to:
[tex]\[
-2x - 12 > -4x
\][/tex]
2. Move all x terms to one side of the inequality:
Add [tex]\(4x\)[/tex] to both sides:
[tex]\[
-2x + 4x - 12 > -4x + 4x
\][/tex]
Which simplifies to:
[tex]\[
2x - 12 > 0
\][/tex]
3. Isolate the x variable:
Add 12 to both sides:
[tex]\[
2x - 12 + 12 > 0 + 12
\][/tex]
This results in:
[tex]\[
2x > 12
\][/tex]
4. Solve for x:
Divide both sides by 2:
[tex]\[
\frac{2x}{2} > \frac{12}{2}
\][/tex]
Which simplifies to:
[tex]\[
x > 6
\][/tex]
### Graphing the Solution on the Number Line
The solution to the inequality [tex]\(x > 6\)[/tex] means all numbers greater than 6. Here is how you can represent this solution on a number line:
1. Draw a number line:
Draw a horizontal line and mark a point at 6 on this line.
2. Open circle at 6:
Place an open circle (not filled) at the point 6 to indicate that 6 is not included in the solution set.
3. Shade to the right of the open circle:
Draw a ray starting at the open circle and extending to the right to indicate all numbers greater than 6 are included in the solution set.
Here is the graph representation:
[tex]\[
\begin{array}{c|ccccccccccc}
\text{Number Line} & \cdots & 4 & 5 & \mathbf{6} & 7 & 8 & 9 & 10 & \cdots \\
\hline
\text{Solution} & \cdots & & & \circ & \rightarrow & & & & \cdots
\end{array}
\][/tex]
### Explanation of the Graph Representation
- The open circle at 6 signifies that 6 is not part of the solution [tex]\( (x > 6) \)[/tex].
- The arrow points right from 6, indicating all numbers greater than 6 are part of the solution set.
So, the solution set for the inequality [tex]\(-2(x + 6) > -4x\)[/tex] is [tex]\( x > 6 \)[/tex], and it is graphically represented on the number line as shown above.
