SECTION 11.7 Systems of Inequalities 837 Figure 42 x y 30 30 20 10 20 10 x 1 y 5 25 x 5 15 y 5 5 (20, 5) (15, 5) (15, 0) (25, 0) (in hundreds) (in hundreds) Financial Planning Raj recently graduated from college and received a signing bonus of $2500 from his employer, which he will invest. As the financial adviser, you recommend that Raj place at least $1500 in Treasury bills yielding 2% and at most $500 in corporate bonds yielding 3%. (a) Using x to denote the amount of money invested in Treasury bills and y to denote the amount invested in corporate bonds, write a system of linear inequalities that describes the possible amounts of each investment. Assume that x and y are in hundreds of dollars. (b) Graph the system. Solution EXAMPLE 12 (a) The system of linear inequalities is x y x y x y 0 0 25 15 5 ≥ ≥ + ≤ ≥ ≤ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ x and y are nonnegative variables since they represent money invested, in hundreds of dollars. The total of the two investments, x y, + cannot exceed $2500. At least $1500 in Treasury bills At most $500 in corporate bonds (b) See the shaded region in Figure 42. Note that the inequalities x 0 ≥ and y 0 ≥ require that the graph of the system be in quadrant I. The graph of the system of linear inequalities in Figure 42 is bounded , because it can be contained within some circle of sufficiently large radius. A graph that cannot be contained in any circle is unbounded . For example, the graph of the system of linear inequalities in Figure 41 is unbounded, since it extends indefinitely in the positive x and positive y directions. Notice in Figures 41 and 42 that those points that belong to the graph and are also points of intersection of boundary lines have been plotted. Such points are referred to as vertices or corner points of the graph. The system graphed in Figure 41 has three corner points: 0, 4 , ( ) 1, 2 , ( ) and 3,0 . ( ) The system graphed in Figure 42 has four corner points: 15, 0 , ( ) 25, 0 , ( ) 20, 5 , ( ) and 15, 5 . ( ) These ideas are used in the next section in developing a method for solving linear programming problems, an important application of linear inequalities. Now Work PROBLEM 45 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 11.7 Assess Your Understanding 1. Solve the inequality: x x 3 4 8 + < − (pp.A80–A81) 2. Graph the equation: x y 3 2 6 − = (pp. 39–40) 3. Graph the equation: x y 9 2 2 + = (pp. 48–51) 4. Graph the equation: y x 4 2 = + (pp. 158–163) 5. True or False The lines x y 2 4 + = and x y 4 2 0 + = are parallel. (p. 40) 6. Solve the inequality: x 4 5 2 − ≤ (pp. 179–181) Concepts and Vocabulary 7. When graphing an inequality in two variables, use if the inequality is strict; if the inequality is nonstrict, use a mark. 8. The graph of a linear equation is a line that separates the xy -plane into two regions called . 9. True or False The graph of a system of inequalities must have an overlapping region. 10. Multiple Choice If the graph of a system of inequalities cannot be contained in any circle, then the graph is: (a) bounded (b) unbounded (c) decomposed (d) composed 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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