Solve the system of inequalities by graphing.
y < 2x + 3
2x - y <= 4
what are the Coordinates
To solve this system of inequalities by graphing, we will graph the two inequalities on the same coordinate plane and find the overlapping region. The coordinates of the overlapping region will be our solution.
First, let's graph the first inequality: y < 2x + 3.
To graph this, we can start with the equation y = 2x + 3 and draw a dashed line. Since it is a strict inequality (<), we will use a dashed line to indicate that the line itself is not included in the solution.
Now, let's choose a test point not on the line. Let's use (0,0). Substitute (0,0) into the inequality y < 2x + 3, we get:
0 < 2(0) + 3
0 < 3
Since the inequality is true, we shade the region below the dashed line.
Next, let's graph the second inequality: 2x - y <= 4.
To graph this, let's start with the equation 2x - y = 4 and draw a solid line. Since it is a non-strict inequality (<=), we will use a solid line to indicate that the line itself is included in the solution.
Let's choose a test point not on the line. Let's use (0,0). Substitute (0,0) into the inequality 2x - y <= 4, we get:
2(0) - (0) <= 4
0 <= 4
Since the inequality is true, we shade the region below the solid line.
Now, let's find the overlapping region of the shaded areas of both inequalities. The overlapping region is the region that satisfies both inequalities.
The overlapping region is the shaded region below the dashed line y < 2x + 3 and the shaded region below the solid line 2x - y <= 4.
The coordinates in the overlapping region are the solution to the system of inequalities. We can see that there is no specific coordinate mentioned in the problem, so the solution is a region.
The overlapping region is the region below both lines, including the area where the lines cross each other.
In the overlapping region, the y-values are less than 2x + 3 and the y-values are greater than or equal to 2x - 4.
Therefore, the solution to the system of inequalities is the shaded region below the dashed line y < 2x + 3 and the shaded region below the solid line 2x - y <= 4.