Let's solve and graph the inequality step-by-step.
### Solving the Inequality
Given inequality:
[tex]\[
-2(x + 6) > -4x
\][/tex]
1. Distribute -2 on the left side
[tex]\[
-2x - 12 > -4x
\][/tex]
2. Add 4x to both sides to combine like terms
[tex]\[
4x - 2x - 12 > -4x + 4x
\][/tex]
[tex]\[
2x - 12 > 0
\][/tex]
3. Add 12 to both sides
[tex]\[
2x - 12 + 12 > 0 + 12
\][/tex]
[tex]\[
2x > 12
\][/tex]
4. Divide by 2
[tex]\[
\frac{2x}{2} > \frac{12}{2}
\][/tex]
[tex]\[
x > 6
\][/tex]
### Solution Set
The solution set for this inequality is [tex]\(x > 6\)[/tex].
### Graphing the Solution on a Number Line
To graph this solution on a number line:
1. Draw a number line with appropriate marks.
2. Locate the point representing [tex]\(6\)[/tex].
3. Place an open circle (since [tex]\(6\)[/tex] is not included in the solution set, indicated by the strict inequality sign [tex]\(>\)[/tex]).
4. Draw a ray starting from just to the right of 6 and extending to the right, indicating all numbers greater than 6.
Here's how it would look on the number line:
[tex]\[
\begin{array}{l}
\begin{tikzpicture}
% Include this package to draw number lines: \usepackage{tikz}
\draw[->] (-2,0) -- (8,0) ; % Draw a number line
\foreach \x in {-1, 0, 1, 2, 3, 4, 5, 6, 7}
\draw (\x,3pt) -- (\x,-3pt)
node[anchor=north] {\x}; % Ticks on the number line
\draw[fill=white] (6,0) circle (3pt); % Open circle at x = 6
\draw[thick,->] (6.1,0) -- (8,0); % Ray starting just right of the open circle going to the right
\end{tikzpicture}
\end{array}
\][/tex]
- The open circle at [tex]\(6\)[/tex] indicates that 6 is not included in the solution.
- The ray extending to the right shows the values for [tex]\(x\)[/tex] that satisfy [tex]\(x > 6\)[/tex].
This represents the solution set for the inequality [tex]\( -2(x+6) > -4x \)[/tex].
