Solve the system of inequalities by graphing.
y < 2x + 3
2x - y <= 4
To graph the system of inequalities, we will first graph both inequalities separately and shade the appropriate regions.
For the first inequality, y < 2x + 3, we will start by graphing the line y = 2x + 3.
To graph this line, we can plot two points and draw a straight line through them. Let's choose x = 0 and x = 2.
When x = 0, y = 2(0) + 3 = 3. So we have the point (0, 3).
When x = 2, y = 2(2) + 3 = 7. So we have the point (2, 7).
Plotting these points and drawing a line through them, we get:
(-3, -3) (0, 3) (2, 7)
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Next, we need to determine which side of the line to shade.
Since y < 2x + 3, we shade the region below the line.
Therefore, for the first inequality, we shade the region below the line.
Now let's move on to the second inequality, 2x - y <= 4.
We will start by graphing the line 2x - y = 4.
To graph this line, we can plot two points and draw a straight line through them. Let's choose x = 0 and x = 4.
When x = 0, 2(0) - y = 4. So y = -4. So we have the point (0, -4).
When x = 4, 2(4) - y = 4. So y = 4. So we have the point (4, 4).
Plotting these points and drawing a line through them, we get:
(-4, 0) (0, -4) (4, 4)
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Next, we need to determine which side of the line to shade.
Since 2x - y <= 4, we shade the region below the line.
Therefore, for the second inequality, we shade the region below the line.
The shaded regions for each inequality are separate and do not overlap.
So when we combine the two shaded regions, the overlapping region is the solution to the system of inequalities.
The solution is the region below the line y = 2x + 3 and below the line 2x - y = 4.
Note: In some cases, the two shaded regions might overlap. In that case, the overlapping region would be the solution. However, in this particular problem, the shaded regions do not overlap.