Could you please help me with this problem? We are to graph each compound inequality.
x ¡Ý -2y and x - 3y < 6
Of course! To graph compound inequalities, we will first graph each inequality separately and then find the overlapping region on the graph.
Let's start with the first inequality: x ≥ -2y.
To graph this inequality, we can start by graphing the line x = -2y. Since it is an "equal to" inequality, we will draw a solid line.
Next, we need to determine which side of the line represents the solution. To do this, we can choose a point not on the line and test it. For example, let's choose the point (0, 0) which is convenient.
Substituting the values into the inequality, we get:
0 ≥ -2(0)
0 ≥ 0
Since 0 is equal to 0, this point satisfies the inequality. Therefore, we shade the region above the line to represent the solutions.
Now let's move on to the second inequality: x - 3y < 6.
To graph this inequality, we can start by graphing the line x - 3y = 6. Since it is a "less than" inequality, we will draw a dashed line.
Again, we choose a point not on the line to test the inequality. Let's use (0, 0) again.
Substituting the values into the inequality, we get:
0 - 3(0) < 6
0 < 6
Since 0 is less than 6, this point satisfies the inequality. Therefore, we shade the region below the line to represent the solutions.
Lastly, we need to find the overlapping region that satisfies both inequalities. This is the region where the shaded regions of both inequalities overlap.
In this case, the overlapping region will be the shaded region above the line x = -2y, and below the dashed line x - 3y = 6.
I hope this helps! Let me know if you have any further questions.