To graph the linear inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex], let's follow a detailed, step-by-step process:
### Step 1: Convert the Inequality to Slope-Intercept Form
We start by converting the inequality to the slope-intercept form, [tex]\(y = mx + b\)[/tex].
1. Given inequality:
[tex]\[
\frac{1}{2} x - 2 y > -6
\][/tex]
2. Isolate the term involving [tex]\(y\)[/tex]. Subtract [tex]\(\frac{1}{2} x\)[/tex] from both sides:
[tex]\[
-2 y > -\frac{1}{2} x - 6
\][/tex]
3. Divide every term by [tex]\(-2\)[/tex]. Remember to reverse the inequality sign when dividing by a negative number:
[tex]\[
y < \frac{1}{4} x + 3
\][/tex]
So, the inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex] is equivalent to [tex]\(y < \frac{1}{4} x + 3\)[/tex].
### Step 2: Graph the Boundary Line
Next, we graph the boundary line, which is given by the equation:
[tex]\[
y = \frac{1}{4} x + 3
\][/tex]
1. Identify the y-intercept ([tex]\(b\)[/tex]): The y-intercept is 3. So, the line crosses the y-axis at (0, 3).
2. Determine the slope ([tex]\(m\)[/tex]): The slope is [tex]\(\frac{1}{4}\)[/tex]. This means for every 4 units increased in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by 1 unit.
3. Plot points using the slope and y-intercept:
- Start at the y-intercept (0, 3).
- From (0, 3), move 4 units to the right and 1 unit up to get the point (4, 4).
- Similarly, move 4 units to the left and 1 unit down from (0, 3) to get the point (-4, 2).
4. Draw a dashed line through these points. The line is dashed because the inequality is strict ([tex]\(<\)[/tex]), meaning the points on the line are not included in the solution set.
### Step 3: Shade the Appropriate Region
The inequality [tex]\(y < \frac{1}{4} x + 3\)[/tex] means we are looking for the region below the line.
1. Pick a test point not on the boundary line (e.g., (0, 0)) and substitute it into the inequality [tex]\(y < \frac{1}{4} x + 3\)[/tex]:
[tex]\[
0 < \frac{1}{4}(0) + 3 \quad \text{which simplifies to} \quad 0 < 3
\][/tex]
2. Since the test point (0, 0) satisfies the inequality, the region below the line is the solution region.
3. Shade the region below the dashed line to represent the solution set of the inequality.
### Final Graph
The final graph includes:
- A dashed line representing [tex]\(y = \frac{1}{4} x + 3\)[/tex].
- The region below this dashed line shaded to indicate all the points that satisfy [tex]\(y < \frac{1}{4} x + 3\)[/tex].
This completes the graph of the linear inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex].
