A system of linear inequalities is shown.

Question

2
x
−
8
y
<
4

x
+
4
y
≥
6

Which graph represents the solution set of the system of inequalities?

Answer 1

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.