Solve the system of inequalities by graphing.
y < 2x + 3
2x - y <= 4
Coordinates
To graph the system of inequalities, we will first graph each inequality separately and then determine the intersection of the shaded regions.
Graph of y < 2x + 3:
To graph y < 2x + 3, start by graphing the equation y = 2x + 3. This is a line with a slope of 2 and a y-intercept of 3. Plot a point at (0, 3), which is the y-intercept, and then use the slope to find more points. For example, if you move 1 unit to the right (change in x = +1), the line will go up by 2 units (change in y = +2). Connect these points with a line.
Next, determine which side of the line to shade by picking a test point not on the line. Let's use (0, 0) as an example. Substitute the x and y values into the inequality: 0 < 2(0) + 3. This simplifies to 0 < 3, which is true. Therefore, shade the region below the line.
Graph of 2x - y <= 4:
To graph 2x - y <= 4, start by graphing the equation 2x - y = 4. This is a line with a slope of 2 and a y-intercept of -4. Plot a point at (0, -4), which is the y-intercept, and then use the slope to find more points. Connect these points with a line.
To determine which side of the line to shade, again pick a test point not on the line. Let's use (0, 0) as an example. Substitute the x and y values into the inequality: 2(0) - (0) <= 4. This simplifies to 0 <= 4, which is true. Therefore, shade the region below the line.
Intersection of the Shaded Regions:
To find the intersection of the shaded regions, look for the overlapping region. This is the region that satisfies both inequalities. In this case, the intersection is the region below both lines.
The coordinates in this region can vary depending on the scale of the graph. Therefore, there are multiple possible sets of coordinates that satisfy the system of inequalities.