To graph the solution to the system of inequalities [tex]\( y \geq -5x - 4 \)[/tex] and [tex]\( y < 2x - 5 \)[/tex], follow these steps:
### 1. Graph the Boundary Lines
First, convert each inequality to an equality:
1. [tex]\( y = -5x - 4 \)[/tex]
2. [tex]\( y = 2x - 5 \)[/tex]
These equations represent the boundary lines for the inequalities.
### 2. Graph the Line [tex]\( y = -5x - 4 \)[/tex]
This line has a slope of -5 and a y-intercept of -4. To plot this:
- Start at the y-intercept (0, -4).
- Use the slope to find another point: from (0, -4), move down 5 units and right 1 unit to get to the point (1, -9).
Plot these points and draw the line. Because the inequality is [tex]\( y \geq -5x - 4 \)[/tex], you will use a solid line to indicate that points on the line satisfy the inequality.
### 3. Graph the Line [tex]\( y = 2x - 5 \)[/tex]
This line has a slope of 2 and a y-intercept of -5. To plot this:
- Start at the y-intercept (0, -5).
- Use the slope to find another point: from (0, -5), move up 2 units and right 1 unit to get to the point (1, -3).
Plot these points and draw the line. Because the inequality is [tex]\( y < 2x - 5 \)[/tex], you will use a dashed line to indicate that points on the line do not satisfy the inequality.
### 4. Determine the Region to Shade
For the inequality [tex]\( y \geq -5x - 4 \)[/tex]:
- Choose a test point not on the line, such as (0, 0).
- Substitute (0, 0) into the inequality: [tex]\( 0 \geq -5(0) - 4 \to 0 \geq -4 \)[/tex] (True).
- Therefore, shade the region above or to the left of the line [tex]\( y = -5x - 4 \)[/tex].
For the inequality [tex]\( y < 2x - 5 \)[/tex]:
- Again, choose a test point not on the line, such as (0, 0).
- Substitute (0, 0) into the inequality: [tex]\( 0 < 2(0) - 5 \to 0 < -5 \)[/tex] (False).
- Therefore, shade the region below or to the right of the line [tex]\( y = 2x - 5 \)[/tex].
### 5. Identify the Solution Region
The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap.
### Summary of Steps
1. Plot the line [tex]\( y = -5x - 4 \)[/tex] as a solid line.
2. Plot the line [tex]\( y = 2x - 5 \)[/tex] as a dashed line.
3. Shade above the solid line for [tex]\( y \geq -5x - 4 \)[/tex].
4. Shade below the dashed line for [tex]\( y < 2x - 5 \)[/tex].
5. The overlapping region is the solution to the system.
### Final Graph
Here is a visual representation of the graph you should obtain:
1. Solid line for [tex]\( y = -5x - 4 \)[/tex] (shaded above).
2. Dashed line for [tex]\( y = 2x - 5 \)[/tex] (shaded below).
3. Overlapping region is the area above the solid line and below the dashed line.
The final region is bounded by these two lines.
This is the region of the coordinate plane that satisfies both [tex]\( y \geq -5x - 4 \)[/tex] and [tex]\( y < 2x - 5 \)[/tex].
