Question
A system of linear inequalities is shown.
2
x
−
8
y
<
4
x
+
4
y
≥
6
Which graph represents the solution set of the system of inequalities?
2
x
−
8
y
<
4
x
+
4
y
≥
6
Which graph represents the solution set of the system of inequalities?
Answers
GPT-4o mini
To graph the solution set of the given system of linear inequalities, we need to analyze each inequality step by step:
1. **First Inequality:**
\[
2x - 8y < 4
\]
We can rewrite this in slope-intercept form (y = mx + b). Start by isolating \(y\):
\[
-8y < -2x + 4
\]
Dividing everything by -8 (and flipping the inequality sign):
\[
y > \frac{1}{4}x - \frac{1}{2}
\]
This defines a region above the line \(y = \frac{1}{4}x - \frac{1}{2}\).
2. **Second Inequality:**
\[
x + 4y \geq 6
\]
Isolate \(y\):
\[
4y \geq -x + 6
\]
Divide by 4:
\[
y \geq -\frac{1}{4}x + \frac{3}{2}
\]
This defines a region above or on the line \(y = -\frac{1}{4}x + \frac{3}{2}\).
### Graphing
Follow these steps to graph the inequalities:
1. **Graph the line** for the first inequality (\(y = \frac{1}{4}x - \frac{1}{2}\)) as a dashed line (since the inequality is <).
- The y-intercept is \(-\frac{1}{2}\), which gives the point (0, -0.5).
- The slope \(\frac{1}{4}\) means moving up 1 unit and right 4 units from that point.
2. **Graph the line** for the second inequality (\(y = -\frac{1}{4}x + \frac{3}{2}\)) as a solid line (since the inequality is ≥).
- The y-intercept is \(\frac{3}{2}\), which gives the point (0, 1.5).
- The slope \(-\frac{1}{4}\) means moving down 1 unit and right 4 units from that point.
3. **Shade the appropriate regions**:
- For the first inequality, shade above the dashed line.
- For the second inequality, shade above and including the solid line.
### Final Graph
The solution set that represents the system of inequalities is the overlapping shaded region above the dashed line (2x - 8y < 4) and above and including the solid line (x + 4y ≥ 6).
### Conclusion
When identifying potential graphs, look for the region that satisfies both inequalities, meaning that it should be in the overlapping area of both shaded regions.
1. **First Inequality:**
\[
2x - 8y < 4
\]
We can rewrite this in slope-intercept form (y = mx + b). Start by isolating \(y\):
\[
-8y < -2x + 4
\]
Dividing everything by -8 (and flipping the inequality sign):
\[
y > \frac{1}{4}x - \frac{1}{2}
\]
This defines a region above the line \(y = \frac{1}{4}x - \frac{1}{2}\).
2. **Second Inequality:**
\[
x + 4y \geq 6
\]
Isolate \(y\):
\[
4y \geq -x + 6
\]
Divide by 4:
\[
y \geq -\frac{1}{4}x + \frac{3}{2}
\]
This defines a region above or on the line \(y = -\frac{1}{4}x + \frac{3}{2}\).
### Graphing
Follow these steps to graph the inequalities:
1. **Graph the line** for the first inequality (\(y = \frac{1}{4}x - \frac{1}{2}\)) as a dashed line (since the inequality is <).
- The y-intercept is \(-\frac{1}{2}\), which gives the point (0, -0.5).
- The slope \(\frac{1}{4}\) means moving up 1 unit and right 4 units from that point.
2. **Graph the line** for the second inequality (\(y = -\frac{1}{4}x + \frac{3}{2}\)) as a solid line (since the inequality is ≥).
- The y-intercept is \(\frac{3}{2}\), which gives the point (0, 1.5).
- The slope \(-\frac{1}{4}\) means moving down 1 unit and right 4 units from that point.
3. **Shade the appropriate regions**:
- For the first inequality, shade above the dashed line.
- For the second inequality, shade above and including the solid line.
### Final Graph
The solution set that represents the system of inequalities is the overlapping shaded region above the dashed line (2x - 8y < 4) and above and including the solid line (x + 4y ≥ 6).
### Conclusion
When identifying potential graphs, look for the region that satisfies both inequalities, meaning that it should be in the overlapping area of both shaded regions.