To graph the linear inequality [tex]\(\frac{1}{2}x - 2y > -6\)[/tex], follow these detailed steps:
1. Start with the inequality:
[tex]\[
\frac{1}{2}x - 2y > -6
\][/tex]
2. Isolate [tex]\( y \)[/tex]:
First, we need to manipulate the inequality to get [tex]\( y \)[/tex] on one side.
Step 1: Move the term involving [tex]\( y \)[/tex] to the other side by adding [tex]\( 6 \)[/tex]:
[tex]\[
\frac{1}{2}x - 2y + 6 > 0
\][/tex]
Step 2: Subtract [tex]\(\frac{1}{2}x\)[/tex] from both sides to isolate the term with [tex]\( y \)[/tex]:
[tex]\[
-2y > -\frac{1}{2}x + 6
\][/tex]
Step 3: Divide both sides by [tex]\(-2\)[/tex]. Remember that dividing or multiplying by a negative number reverses the inequality sign:
[tex]\[
y < \frac{1}{4}x - 3
\][/tex]
3. Graph the boundary line:
The inequality [tex]\( y < \frac{1}{4}x - 3 \)[/tex] corresponds to a boundary line given by the equation [tex]\( y = \frac{1}{4}x - 3 \)[/tex].
- Plot the line [tex]\( y = \frac{1}{4}x - 3 \)[/tex].
- Since the original inequality is a strict inequality ( [tex]\( < \)[/tex] and not [tex]\( \leq \)[/tex]), the line will be dashed, indicating that points on the line are not included in the solution.
4. Identify the region to shade:
After plotting the dashed boundary line, determine which side of the line represents the solution to the inequality.
- Pick a test point that is not on the line, such as the origin [tex]\((0, 0)\)[/tex].
- Substitute [tex]\((0, 0)\)[/tex] into the inequality [tex]\( y < \frac{1}{4}(0) - 3 \)[/tex]:
[tex]\[
0 < -3
\][/tex]
This statement is false, so the region that does not include the origin is the solution region.
- Thus, you should shade the region below the dashed line [tex]\( y = \frac{1}{4}x - 3 \)[/tex].
By following these steps, the graph of the linear inequality [tex]\(\frac{1}{2}x-2y > -6\)[/tex] is represented by the region below the dashed line [tex]\( y < \frac{1}{4}x - 3 \)[/tex].
