To solve the given system of inequalities graphically, we need to plot each inequality on a coordinate system and identify the overlapping region that satisfies both inequalities.
Step 1: Graph the first inequality
- Inequality: \( y \geq 3x - 6 \)
This inequality represents a line with a slope of 3 and a y-intercept of -6. Since it's "greater than or equal to," we will draw a solid line for the equation \( y = 3x - 6 \) and shade the area above this line.
- Y-intercept: At \( x = 0 \), \( y = 3(0) - 6 = -6 \). So the point (0, -6) is on the line.
- Another point: At \( x = 2 \), \( y = 3(2) - 6 = 0 \). So the point (2, 0) is another point on the line.
Step 2: Graph the second inequality
- Inequality: \( y < -2x - 1 \)
This inequality represents a line with a slope of -2 and a y-intercept of -1. Here, we will draw a dashed line for the equation \( y = -2x - 1 \) and shade the area below this line.
- Y-intercept: At \( x = 0 \), \( y = -2(0) - 1 = -1 \). So the point (0, -1) is on the line.
- Another point: At \( x = 1 \), \( y = -2(1) - 1 = -3 \). So the point (1, -3) is another point on the line.
Step 3: Identify the solution region
Once both lines are plotted, we need to look for the region where the shaded areas overlap:
- The area above the solid line for the first inequality.
- The area below the dashed line for the second inequality.
The intersection of these shaded regions is the solution set for the system of inequalities.
Step 4: Find a point in the solution set
To find a point that belongs to this overlapping region, we can test a few values.
For example, consider the point \( (1, 0) \):
-
For the first inequality: \[ 0 \geq 3(1) - 6 \rightarrow 0 \geq -3 \quad \text{(True)} \]
-
For the second inequality: \[ 0 < -2(1) - 1 \rightarrow 0 < -3 \quad \text{(False)} \]
Now let's try \( (0, 0) \):
-
For the first inequality: \[ 0 \geq 3(0) - 6 \rightarrow 0 \geq -6 \quad \text{(True)} \]
-
For the second inequality: \[ 0 < -2(0) - 1 \rightarrow 0 < -1 \quad \text{(False)} \]
Now let's try \( (-1, 1) \):
-
For the first inequality: \[ 1 \geq 3(-1) - 6 \rightarrow 1 \geq -3 - 6 = -9 \quad \text{(True)} \]
-
For the second inequality: \[ 1 < -2(-1) - 1 \rightarrow 1 < 2 - 1 \rightarrow 1 < 1 \quad \text{(False)} \]
Now let's try \( (-3, 0) \):
-
For the first inequality: \[ 0 \geq 3(-3) - 6 \rightarrow 0 \geq -9 \quad \text{(True)} \]
-
For the second inequality: \[ 0 < -2(-3) - 1 \rightarrow 0 < 6 - 1 \rightarrow 0 < 5 \quad \text{(True)} \]
Thus, the point \((-3, 0)\) is a solution that lies in the overlapping area of the shaded regions.
Conclusion
The coordinates of a point in the solution set are \((-3, 0)\).
1 answer