Question: Write the system of linear inequalities that defines the shaded region.
A ) - x + y > 0 and x + y > 0
B ) - y > 0 and x > 0
C ) - x > 0 and y > 0
D ) None of the above
Steps to derive
1 There are two lines that bound the shaded region. So, the system must have two linear inequalities.
2 Line (1) passes through the points (-1, -1) and (1, 1).
3 Slope of a line m = (y2-y1)(x2-x1)
[Formula for slope.]
4 m = (1-(-1))(1-(-1)) = 1
[Substitute coordinates in the formula.]
5 Since Line (1) crosses y-axis at the point (0, 0), its equation can be found using the slope-intercept form.
6 y = mx + b
[Write slope-intercept form.]
7 - x + y = 0
[Substitute 1 for m and 0 for b and simplify.]
8 Since the shaded region is above the dashed boundary line (1), the inequality is
- x + y > 0.
9 Line (2) passes through the points (1, -1) and (-1, 1).
10 m = (1-(-1))(-1-1) = -1
[Substitute the coordinates in the slope formula.]
11 Since Line (2) crosses y-axis at the point (0, 0), its equation can be found using the slope-intercept form.
12 y = mx + b
[Write slope-intercept form.]
13 x + y = 0
[Substitute -1 for m and 0 for b and simplify.]
14 Since the shaded region is above the dashed boundary line (2), the inequality is
x + y > 0.
15 The system of inequalities that describes the shaded region is - x + y > 0 and x + y > 0.
Hence the right answer is Option A
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