To graph the system
y ≤ 2x + 1
y < -x - 1
first, you have to graph the lines
y = 2x + 1
y = -x - 1
From the y-intercept and the slope, we know that the first line passes through the points:
(0, 1)
(0+1, 1+2) = (1, 3)
From the y-intercept and the slope, we know that the second line passes through the points:
(0, -1)
(0+1, -1-1) = (1, -2)
Next, we have to find if a point satisfies each equation or not, in order to know which region we have to shade. Taking for example the point (0, 0) and replacing it into the first inequality,
0 ≤ 2(0) + 1
0 ≤ 1
which is true, then we have to shade the region below the line y = 2x + 1
Replacing (0, 0) into the second inequality,
0 < -0 - 1
0 < -1
which is false, then we have to shade the region below the dotted line y = -x - 1
The final result is shown in the next picture.
This corresponds to graph X
