SECTION 11.7 Systems of Inequalities 835 Solution First, graph the lines x y Y x 2 2 1 ( ) + = = − + and x y Y x 2 4 2 4 . 2 ( ) − = = − See Figure 35. Notice that the graphs divide the viewing window into four regions. Select a test point for each region, and determine whether the point makes both inequalities true. We choose to test 0, 0 , 2, 3 , 4, 0 , ( ) ( ) ( ) and 2, 2. ( ) − Figure 36(a) shows that 2, 3 ( ) makes both inequalities true. The graph of the system of inequalities is shown in Figure 36(b) using a TI-84 Plus CE and in Figure 36(c) using Desmos. Figure 35 5 Y2 5 2x 2 4 Y1 5 2x 1 2 25 26 10 (b) 5 (2, 3) (4, 0) (0, 0) (2, 22) 25 26 10 (c) Figure 36 (a) Rather than testing four points, we could test just point 0, 0 ( ) on each inequality. For example, 0,0 ( ) does not satisfy x y 2, + ≥ so shade above the line x y 2. + = In addition, 0, 0 ( ) does satisfy x y 2 4, − ≤ so shade above the line x y 2 4. − = The intersection of the shaded regions gives the result presented in Figure 36(b). Now Work PROBLEM 23 Graphing a System of Linear Inequalities by Hand Graph the system: x y x y 2 0 + ≤ + ≥ ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ Solution EXAMPLE 8 See Figure 37. The overlapping purple-shaded region between the two boundary lines is the graph of the system. Figure 37 x 1 y # 2 x 1 y $ 0 Graph of x y 23 3 23 3 x 1 y 5 2 x 1 y 5 0 Now Work PROBLEM 29 Graphing a System of Linear Inequalities by Hand Graph the systems: (a) x y x y 2 0 2 2 − ≥ − ≥ ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ (b) x y x y 2 2 2 6 + ≤ + ≥ ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ EXAMPLE 9 (continued)
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