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The system of linear inequalities is:
y ≥ 2x + 1
y ≤ 2x – 2
To graph the solution to this system, we can begin by graphing the boundary lines for each inequality.
For the first inequality, we can draw a line with slope 2 passing through the point (0, 1) since the y-intercept is (0,1).
For the second inequality, we can draw a line with slope 2 passing through the point (0, -2) since the y-intercept is (0, -2).
Once we have graphed the boundary lines, we need to determine which side of each line satisfies the corresponding inequality.
For the first inequality, we test a point not on the line, such as (0, 0), and substitute the coordinates into the inequality:
y ≥ 2x + 1 0 ≥ 2(0) + 1 0 ≥ 1
Since 0 is not greater than or equal to 1, we shade the region above the line to indicate that it does not satisfy the inequality.
For the second inequality, we test the same point, (0, 0), and substitute the coordinates into the inequality:
y ≤ 2x – 2 0 ≤ 2(0) – 2 0 ≤ -2
Since 0 is not less than or equal to -2, we shade the region below the line to indicate that it does not satisfy the inequality.
The shaded regions for both inequalities do not overlap, indicating that there is no point that satisfies both inequalities. Therefore, there is no solution to the system of linear inequalities.
The graph of the solution is shown below:
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