To graph the inequality [tex]\(x - 2y \geq -12\)[/tex], follow these steps:
1. Rewrite the Inequality:
First, we need to rewrite the inequality in a form that can be easily graphed.
Given:
[tex]\[
x - 2y \geq -12
\][/tex]
Isolate [tex]\(y\)[/tex] on one side of the inequality:
[tex]\[
x - 2y \geq -12
\][/tex]
Rearrange it to solve for [tex]\(y\)[/tex]:
[tex]\[
x - 2y \geq -12
\][/tex]
Add 12 to both sides:
[tex]\[
x - 2y + 12 \geq 0
\][/tex]
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
-2y \geq -x + 12
\][/tex]
Divide each term by [tex]\(-2\)[/tex]. Remember, dividing by a negative number reverses the inequality:
[tex]\[
y \leq \frac{1}{2}x - 6
\][/tex]
2. Graph the Boundary Line:
The boundary line is given by:
[tex]\[
y = \frac{1}{2}x - 6
\][/tex]
This boundary line should be solid because the inequality is [tex]\(\leq\)[/tex] (not just [tex]\(<\)[/tex]).
To plot this line:
- Find the intercepts:
- When [tex]\(x = 0\)[/tex]:
[tex]\[
y = \frac{1}{2}(0) - 6 = -6
\][/tex]
So, the y-intercept is [tex]\((0, -6)\)[/tex].
- When [tex]\(y = 0\)[/tex]:
[tex]\[
0 = \frac{1}{2}x - 6 \implies \frac{1}{2}x = 6 \implies x = 12
\][/tex]
So, the x-intercept is [tex]\((12, 0)\)[/tex].
- Plot these points: [tex]\((0, -6)\)[/tex] and [tex]\((12, 0)\)[/tex].
- Draw a straight line through these points.
3. Shading the Solution Region:
Since the inequality is [tex]\(y \leq \frac{1}{2}x - 6\)[/tex], shade the region below the line, as this includes all points where [tex]\(y\)[/tex] is less than or equal to [tex]\(\frac{1}{2}x - 6\)[/tex].
In conclusion, the graph of the inequality [tex]\(x - 2y \geq -12\)[/tex] includes the boundary line [tex]\(y = \frac{1}{2}x - 6\)[/tex] (drawn as a solid line), with the region below this line shaded, representing the solutions to the inequality.
