To graph the linear inequality [tex]\( \frac{1}{2} x - 2 y > -6 \)[/tex], follow these steps:
1. Rewrite the Inequality in Slope-Intercept Form:
First, we'll rewrite the given inequality in the form [tex]\( y < mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Starting with the given inequality:
[tex]\[
\frac{1}{2} x - 2 y > -6
\][/tex]
Isolate [tex]\( y \)[/tex] on one side of the inequality:
[tex]\[
-2 y > -\frac{1}{2} x - 6
\][/tex]
Divide every term by [tex]\(-2\)[/tex] and remember to reverse the inequality sign because we are dividing by a negative number:
[tex]\[
y < \frac{x}{4} + 3
\][/tex]
Now, the inequality is in the form [tex]\( y < \frac{1}{4} x + 3 \)[/tex].
2. Graph the Boundary Line:
The boundary of the inequality [tex]\( y < \frac{1}{4} x + 3 \)[/tex] is the line [tex]\( y = \frac{1}{4} x + 3 \)[/tex]. This line has a slope of [tex]\(\frac{1}{4}\)[/tex] and a y-intercept of 3.
- To plot the boundary line, start by plotting the y-intercept (0, 3) on the coordinate plane.
- From this point, use the slope to find another point on the line. Since the slope is [tex]\(\frac{1}{4}\)[/tex], you can go up 1 unit and right 4 units from the y-intercept. This gets you to the point (4, 4).
- Draw a dashed line through these points. The line should be dashed because the inequality is strictly less than ([tex]\(<\)[/tex]), not less than or equal to ([tex]\(\leq\)[/tex]).
3. Shade the Appropriate Region:
Since the inequality is [tex]\( y < \frac{1}{4} x + 3 \)[/tex], you need to shade the region below the dashed line. This is the set of points where the y-values are less than the values on the line [tex]\( y = \frac{1}{4} x + 3 \)[/tex].
4. Check Your Solution:
- Select a test point not on the boundary line, like (0,0), to verify the correct region to shade.
[tex]\[
\text{Substitute } (0, 0) \text{ into } \frac{1}{2} x - 2 y > -6
\][/tex]
[tex]\[
\frac{1}{2}(0) - 2(0) > -6
\][/tex]
[tex]\[
0 > -6
\][/tex]
This is true, so the region containing (0,0) is the solution. Thus, you shade this region.
By following these steps, you have successfully graphed the inequality [tex]\( \frac{1}{2} x - 2 y > -6 \)[/tex], which translates to [tex]\( y < \frac{x}{4} + 3 \)[/tex] on the coordinate plane.
