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Question: Write a system of linear inequalities that define the shaded region.



A ) x + y = -1 and y = 1
B ) x + y ≥ -1 and x + y < 1
C ) x ≥ -1 and x + y = 1
D ) None of the above

Steps to derive

1 The shaded region is between the two lines. So, the system must have two linear inequalities.

2 Line (1) passes through the points (0, -1) and (-1, 0).

 [Use slope formula to find slope of line (1).]
3 m = (y2 -y1)(x2 -x1)

 [Substitute coordinates in the formula and simplify.]
4 m = (0-(-1))(-1-0) = -1

5 Since, line (1) crosses y-axis at the point (0, -1), its equation can be found using the slope-intercept form.
 [The y intercept of the line is -1]

 [Write slope-intercept form.]
6 y = mx + b

 [Substitute -1 for m and -1 for b and simplify.]
7 x + y = -1

8 Since, the shaded region is on and above the solid boundary line (1), the inequality is x + y ≥ -1.

9 Line (2) passes through the points (1, 0) and (0, 1).

 [Substitute coordinates in the slope formula and simplify.]
10 m = (1-0)(0-1) = -1

11 Since, line (2) crosses y-axis at the point (0, 1), its equation can be found using the slope-intercept form.
 [The y-intercept of the line is 1.]

 [Substitute -1 for m and 1 for b in the slope intercept form, y = mx + b.]
12 x + y = 1

13 Since, the shaded region is below the dashed boundary line (2), the inequality is x + y < 1.

14 The system of inequalities that describes the shaded region is x + y ≥ -1 and x + y < 1.



Hence the right answer is Option B

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