Section 3.5 Linear Inequalities and Systems of Linear Inequalities in Two Variables 275 3. Graphing a Feasible Region When two variables are related under certain constraints, a system of linear inequalities can be used to show a region of feasible values for the variables. The feasible region represents the ordered pairs that are true for each inequality in the system. Graphing a Feasible Region Example 7 Susan has two tests on Friday: one in chemistry and one in psychology. Because the two classes meet in consecutive hours, she has no study time between tests. Susan estimates that she has a maximum of 12 hr of study time before the tests, and she must divide her time between chemistry and psychology. Let x represent the number of hours Susan spends studying chemistry. Let y represent the number of hours Susan spends studying psychology. a. Find a set of inequalities to describe the constraints on Susan’s study time. b. Graph the constraints to find the feasible region defining Susan’s study time. Solution: a. Because Susan cannot study chemistry or psychology for a negative period of time, we have x 0 and y 0. Furthermore, her total time studying cannot exceed 12 hr: x y 12. A system of inequalities that defines the constraints on Susan’s study time is: x 0 y 0 x y 12 b. The first two conditions and x 0 y 0 represent the set of points in the first quadrant. The third condition x y 12 represents the set of points below and including the line (Figure 3-21). x y 12 16 14 12 10 8 6 4 2 Discussion: 1. Refer to the feasible region. Is the ordered pair (8, 5) part of the feasible region? No. The ordered pair (8, 5) indicates that Susan spent 8 hr studying chemistry and 5 hr studying psychology. This is a total of 13 hr, which exceeds the constraint that Susan only had 12 hr to study. The point (8, 5) lies outside the feasible region, above the line x y 12 (Figure 3-22). 2. Is the ordered pair (7, 3) part of the feasible region? Yes. The ordered pair (7, 3) indicates that Susan spent 7 hr studying chemistry and 3 hr studying psychology. This point lies within the feasible region and satisfies all three constraints. True True True x 0 7 0 y 0 3 0 x y 12 172 132 12 Notice that the ordered pair (7, 3) corresponds to a point where Susan is not making full use of the 12 hr of study time. y Hours (chemistry) Hours (psychology) x x y 12 42 2 4 6 8 10 12 14 16 2 4 y x y 12 Hours (chemistry) Hours (psychology) x 16 14 42 2 4 6 8 10 12 14 16 2 4 (8, 5) (7, 3) 12 10 8 6 4 2 Figure 3-21 Figure 3-22
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