Graph the system of inequalities (shade each half-plane solution,not only the overall solution).
y<= 3/2x+3
2x + y<10
y>-1
http://www.wolframalpha.com/input/?i=plot+y%3C%3D+3%2F2+x%2B3,+2x+%2B+y%3C10,+y%3E-1
To graph the system of inequalities, we will plot the individual inequalities on a coordinate plane and shade the regions that satisfy each inequality.
1. Graph the inequality y <= (3/2)x + 3:
- Start by graphing the line y = (3/2)x + 3.
- Find two points on this line by assigning values to 'x' and calculating the corresponding 'y' values.
Let's assume x = 0, then y = (3/2)(0) + 3 = 3.
And if x = 2, then y = (3/2)(2) + 3 = 6.
- Plot the two points (0, 3) and (2, 6) on the coordinate plane. Then, connect them with a straight line.
- To determine which region to shade, select a point not on the line. The origin, (0,0), is a convenient choice.
- Substitute the coordinates (x,y) = (0,0) into the inequality y <= (3/2)x + 3.
We have 0 <= (3/2)(0) + 3, which simplifies to 0 <= 3. Since this is true, shade the region below the line.
2. Graph the inequality 2x + y < 10:
- Start by graphing the line 2x + y = 10.
- Find two points on this line by assigning values to 'x' and calculating the corresponding 'y' values.
Let's assume x = 0, then y = 10 - 2(0) = 10.
And if x = 5, then y = 10 - 2(5) = 0.
- Plot the two points (0, 10) and (5, 0) on the coordinate plane. Then, connect them with a straight line.
- To determine which region to shade, select a point not on the line. Again, the origin (0,0) is a convenient choice.
- Substitute the coordinates (x,y) = (0,0) into the inequality 2x + y < 10.
We have 2(0) + 0 < 10, which simplifies to 0 < 10. Since this is true, shade the region below the line.
3. Graph the inequality y > -1:
- Start by graphing the line y = -1.
- Plot a point on this line, which is (-1,-1).
- Draw a horizontal dashed line passing through this point.
- To determine which region to shade, select a point not on the line. For simplicity, choose the point (0,0).
- Substitute the coordinates (0,0) into the inequality y > -1. Since 0 is greater than -1, shade the region above the line.
After graphing all three inequalities, the shaded regions will overlap to form the solution set of the system of inequalities.