836 CHAPTER 11 Systems of Equations and Inequalities Solution (a) See Figure 38. The overlapping purple-shaded region is the graph of the system. Note that the graph of the system is identical to the graph of the single inequality x y 2 2. − ≥ (b) See Figure 39. Here, because no overlapping region results, there are no points in the xy-plane that simultaneously satisfy each inequality. The system has no solution. Figure 38 2x 2 y $ 0 2x 2 y $ 2 Graph of x y 23 3 23 3 2x 2 y 5 2 2x 2 y 5 0 Figure 39 y 24 4 24 6 x x 1 2y 5 6 x 1 2y 5 2 Graphing a System of Nonlinear Inequalities by Hand Graph the region below the graph of x y 2 + = and above the graph of y x 4 2 = − by graphing the system y x x y 4 2 2 ≥ − + ≤ ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ Label all points of intersection. Solution EXAMPLE 10 Figure 40 shows the graph of the region above the graph of the parabola y x 4 2 = − and below the graph of the line x y 2. + = The points of intersection are found by solving the system of equations y x x y 4 2 2 = − + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ Use substitution to find x x x x x x x x 4 2 6 0 3 2 0 3 or 2 2 2 ( ) ( )( ) + − = + − = + − = = − = The two points of intersection are 3, 5 ( ) − and 2,0 . ( ) Figure 40 x y –4 4 –4 x 1 y 5 2 y 5 x 2 2 4 (–3, 5) 4 (2, 0) Now Work PROBLEM 37 Graphing a System of Four Linear Inequalities by Hand Graph the system: x y x y x y 3 2 4 0 0 + ≥ + ≥ ≥ ≥ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ EXAMPLE 11 Solution See Figure 41. The two inequalities x 0 ≥ and y 0 ≥ require the graph of the system to be in quadrant I. Concentrate on the remaining two inequalities. The intersection of the graphs of these two inequalities and quadrant I is shown in dark purple. Figure 41 21 21 x 5 1 x 1 y 5 3 2x 1 y 5 4 5 1 solution (1, 2) (0, 4) (3, 0) y
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