To graph the system of inequalities \( 4x - y \geq 4 \) and \( y < 2 \), let's follow a detailed step-by-step approach.
### Step 1: Rewrite the Inequalities
1. For the first inequality \( 4x - y \geq 4 \):
- Rearrange it to solve for \( y \):
[tex]\[
4x - y \geq 4 \implies y \leq 4x - 4
\][/tex]
- This represents a line \( y = 4x - 4 \) with a shaded region below or on the line.
2. For the second inequality \( y < 2 \):
- This inequality is already solved for \( y \):
[tex]\[
y < 2
\][/tex]
- This represents a horizontal line \( y = 2 \) with a shaded region below the line.
### Step 2: Identify the Boundary Lines
1. Boundary for the first inequality:
- Line equation: \( y = 4x - 4 \).
2. Boundary for the second inequality:
- Line equation: \( y = 2 \).
### Step 3: Graph the Boundary Lines
1. Graph \( y = 4x - 4 \):
- To plot this line, find two points:
- When \( x = 0 \):
[tex]\[
y = 4(0) - 4 = -4 \implies (0, -4)
\][/tex]
- When \( x = 1 \):
[tex]\[
y = 4(1) - 4 = 0 \implies (1, 0)
\][/tex]
- Draw the line through these points. Since the inequality is \( \geq \), the line will be solid.
2. Graph \( y = 2 \):
- This is a horizontal line passing through \( y = 2 \).
- Draw a dashed line (since the inequality is \( < \)) for \( y = 2 \).
### Step 4: Shade the Feasible Region
1. For \( y \leq 4x - 4 \):
- Shade the region below or on the solid line \( y = 4x - 4 \).
2. For \( y < 2 \):
- Shade the region below the dashed line \( y = 2 \).
### Step 5: Determine the Intersection
- The solution to the system of inequalities is the region where the shaded areas overlap. This gives us the final feasible region.
### Step 6: Final Graph
- The intersection of the two shaded regions is the feasible region bounded by \( y = 4x - 4 \) and below \( y = 2 \).
### Summary of the Boundary Equations
- \( 4x - y - 4 = 0 \) which simplifies to \( y = 4x - 4 \).
- \( y = 2 \).
By following these steps, you should be able to graph the system of inequalities [tex]\( 4 x - y \geq 4 \text{ and } y < 2 \)[/tex] and identify the feasible region which satisfies both inequalities.
