2x − y > 4
x + y < −1
To solve this system of inequalities, we can graph each one separately on a coordinate plane and then look for the overlapping region.
First, let's graph 2x - y > 4:
We can rearrange this inequality to isolate y:
- y > -2x + 4
y < 2x - 4
Now we can graph the line y = 2x - 4 (dashed because it is less than) and shade the region below it:
Next, let's graph x + y < -1:
We can rearrange this inequality to isolate y:
y < -x - 1
Now we can graph the line y = -x - 1 (dashed because it is less than) and shade the region below it:
Finally, we can see that the overlapping region of the shaded regions is the solution to the system of inequalities:
Thus, the solution to the system is the shaded triangle with vertices (-3, -5), (-3, -2), and (1, -2).
To solve the system of inequalities:
1. Begin by solving the second inequality for y:
x + y < -1
y < -x - 1
2. Since we have a strict inequality symbol, we need to represent the inequality as a dashed line on a graph. Plot the line y = -x - 1 with a dashed line.
3. Next, solve the first inequality for y:
2x - y > 4
-y > -2x + 4
y < 2x - 4
4. Again, represent the inequality as a dashed line on the graph. Plot the line y = 2x - 4 with a dashed line.
5. Now, shade the appropriate region to represent the solutions to both inequalities. Since we want the values that satisfy both inequalities, shade the region below the line y = -x - 1 and above the line y = 2x - 4.
6. The overlapping shaded region represents the solutions to the system of inequalities.
Graphically solving the system of inequalities produces the solution.