To graph the linear inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex], we can follow these steps:
### Step 1: Convert the Inequality to an Equation
First, we convert the inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex] into an equation for easier graphing. The corresponding equation is:
[tex]\[
\frac{1}{2} x - 2 y = -6
\][/tex]
### Step 2: Solve for [tex]\(y\)[/tex]
To place the equation into slope-intercept form [tex]\(y = mx + b\)[/tex], we solve for [tex]\(y\)[/tex].
Starting with:
[tex]\[
\frac{1}{2} x - 2 y = -6
\][/tex]
Isolate [tex]\(y\)[/tex]:
1. Add [tex]\(2y\)[/tex] to both sides:
[tex]\[
\frac{1}{2} x = 2 y - 6
\][/tex]
2. Add 6 to both sides:
[tex]\[
\frac{1}{2} x + 6 = 2y
\][/tex]
3. Divide every term by 2:
[tex]\[
\left(\frac{1}{2}\right) \frac{x}{2} + \frac{6}{2} = y
\][/tex]
Simplifying, we get:
[tex]\[
y = \frac{1}{4} x + 3
\][/tex]
### Step 3: Graph the Equation
Plot the line [tex]\(y = \frac{1}{4} x + 3\)[/tex].
1. Start at the y-intercept (0, 3).
2. Use the slope [tex]\(\frac{1}{4}\)[/tex] to determine another point on the line. From (0, 3), go up 1 unit and right 4 units to reach another point, (4, 4).
### Step 4: Determine the Shading Region
The original inequality is [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex]. Since it is a strict inequality ([tex]\(>\)[/tex]), we will draw a dashed line to represent [tex]\(y = \frac{1}{4} x + 3\)[/tex].
To determine which side of the line to shade, pick a test point that is not on the line. A convenient choice is the origin (0, 0).
Substitute (0, 0) into the inequality:
[tex]\[
\frac{1}{2}(0) - 2(0) > -6
\][/tex]
[tex]\[
0 > -6
\][/tex]
This statement is true, so the region that includes the origin is the solution region. Therefore, we shade the area above the dashed line.
### Conclusion
The graph of the linear inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex] is a dashed line representing the equation [tex]\(y = \frac{1}{4} x + 3\)[/tex], and the region above this line is shaded.
