To graph the system of inequalities [tex]\(y \geq \frac{4}{5} x - \frac{1}{5}\)[/tex] and [tex]\(y \leq 2x + 6\)[/tex], we need to follow these steps:
1. Graph the boundary lines:
- First, graph the line [tex]\(y = \frac{4}{5} x - \frac{1}{5}\)[/tex].
- Next, graph the line [tex]\(y = 2x + 6\)[/tex].
2. Determine the shading for each inequality:
- For [tex]\(y \geq \frac{4}{5} x - \frac{1}{5}\)[/tex], shade the region above the line [tex]\(y = \frac{4}{5} x - \frac{1}{5}\)[/tex].
- For [tex]\(y \leq 2x + 6\)[/tex], shade the region below the line [tex]\(y = 2x + 6\)[/tex].
### Step-by-Step Process:
1. Graphing the line [tex]\(y = \frac{4}{5} x - \frac{1}{5}\)[/tex]:
- Determine two points to plot the line:
- When [tex]\(x = 0\)[/tex]: [tex]\(y = \frac{4}{5} \cdot 0 - \frac{1}{5} = -\frac{1}{5}\)[/tex]. So one point is (0, -0.2).
- When [tex]\(x = 5\)[/tex]: [tex]\(y = \frac{4}{5} \cdot 5 - \frac{1}{5} = 4 - 0.2 = 3.8\)[/tex]. So another point is (5, 3.8).
- Draw the line through these two points.
- Since the inequality is [tex]\(y \geq \frac{4}{5} x - \frac{1}{5}\)[/tex], use a solid line and shade the region above it.
2. Graphing the line [tex]\(y = 2x + 6\)[/tex]:
- Determine two points to plot the line:
- When [tex]\(x = 0\)[/tex]: [tex]\(y = 6\)[/tex]. So one point is (0, 6).
- When [tex]\(x = -3\)[/tex]: [tex]\(y = 2 \cdot (-3) + 6 = -6 + 6 = 0\)[/tex]. So another point is (-3, 0).
- Draw the line through these two points.
- Since the inequality is [tex]\(y \leq 2x + 6\)[/tex], use a solid line and shade the region below it.
3. Finding the feasible region:
- The feasible region is where the shaded areas of both inequalities overlap.
- This is the region above the line [tex]\(y = \frac{4}{5} x - \frac{1}{5}\)[/tex] and below the line [tex]\(y = 2x + 6\)[/tex].
4. Plotting the solution:
- Your final graph should show:
- Both lines with appropriate shading.
- The region where both shaded areas overlap is the solution to the system.
### Summary Diagram:
To help you visualize:
1. Draw the line [tex]\(y = \frac{4}{5} x - \frac{1}{5}\)[/tex], use solid line, and shade above.
2. Draw the line [tex]\(y = 2x + 6\)[/tex], use solid line, and shade below.
3. The solution is the overlapping shaded region.
This graph represents the solution to the system of inequalities [tex]\(y \geq \frac{4}{5} x - \frac{1}{5}\)[/tex] and [tex]\(y \leq 2x + 6\)[/tex].
