Graph the solution set for the system of linear inequalities y<2x+3 and y>=2x-6

It is a band of constant width going up to the right with slope = 2 starting from between the points (0,-6) and (0,3)

To graph the solution set for the system of linear inequalities y < 2x + 3 and y >= 2x - 6, follow these steps:

Step 1: Graph the line y = 2x + 3.
To graph this line, start by plotting the y-intercept at (0, 3). Then, use the slope (which is 2 in this case) to find additional points. Since the slope is positive, you can go up 2 units and over 1 unit to graph the next point. Connect the plotted points to form a straight line.

Step 2: Graph the line y = 2x - 6.
Using the same process as in Step 1, plot the y-intercept at (0, -6) and find additional points by using the slope of 2. Connect the points to form another straight line.

Step 3: Identify the solution region.
To find the solution region, look at the shading between the lines.

Since y < 2x + 3, the solution lies below the line y = 2x + 3. Therefore, shade the region below the line.

On the other hand, y >= 2x - 6, so the solution lies above or on the line y = 2x - 6. Therefore, do not shade the region below this line.

Step 4: Find the overlapping region.
The overlapping region between the shaded area below the line y = 2x + 3 and the non-shaded region above or on the line y = 2x - 6 represents the solution set for the system of linear inequalities.

This overlapping region is the solution set of the system.

Note: The solution is the interior of the shaded region (excluding the lines) and any points on the lines that satisfy both inequalities.

To see a visual representation of this solution set, you can refer to the attached graph.

To graph the solution set for the system of linear inequalities y < 2x + 3 and y >= 2x - 6, we will start by graphing each inequality separately and then shade the overlapping region.

Step 1: Graph the inequality y < 2x + 3:
To graph this inequality, we will treat it as an equation and consider it as a boundary line. We can choose any arbitrary value for x and substitute it into the equation to find the corresponding y-coordinate. Repeat this process with multiple x-values to obtain several points on the line.

For example, let's choose x = 0:
y = 2(0) + 3
y = 3

One point on the line is (0, 3). Similarly, let's choose x = 1:
y = 2(1) + 3
y = 5

Another point on the line is (1, 5).

Repeat this process with more x-values and plot the corresponding points on the graph. Then draw a line passing through the plotted points. However, since the inequality is y < 2x + 3 (not including equals to), we will use a dashed line to represent the inequality.

Step 2: Graph the inequality y >= 2x - 6:
Similarly, we will treat this inequality as an equation and find the points on the line. Let's choose x = 0:
y = 2(0) - 6
y = -6

The point (0, -6) is on the line. Choosing x = 1:
y = 2(1) - 6
y = -4

Another point on the line is (1, -4).

Repeat this process and plot the points on the graph and draw a solid line since the inequality is y >= 2x - 6 (including equality).

Step 3: Shade the overlapping region:
Now that we have both lines graphed, we need to shade the region where both inequalities are satisfied. In this case, we see that both lines intersect in the area where y is greater than or equal to the line y = 2x - 6 and less than the line y = 2x + 3.

Therefore, shade the region between the two lines, including the solid line y >= 2x - 6, but excluding the dashed line y < 2x + 3.

The shaded region represents the solution set for the given system of linear inequalities.