834 CHAPTER 11 Systems of Equations and Inequalities NOTE A second approach that could be used in Example 5 is to solve the inequality + ≤ x y 3 6 for y, ≤− + y x 3 6, graph the corresponding line =− + y x3 6, and then shade below since the inequality is of the form <y or ≤y (shade above if of the form >y or ≥y ). j point 1, 2 ( ) − that is below Y .1 Figure 33(b) shows the graph of the inequality using a TI-84 Plus CE. Figure 32 + = x y 3 6 8 Y1 5 23x 1 6 22 26 10 (b) 8 (5, 5) (21, 2) 22 26 10 Figure 33 + ≤ x y 3 6 (a) Now Work PROBLEM 15 USING A GRAPHING UTILITY 3 Graph a System of Inequalities The graph of a system of inequalities in two variables x and y is the set of all points x y , ( ) that simultaneously satisfy each inequality in the system.The graph of a system of inequalities can be obtained by graphing each inequality individually and then determining where, if at all, they intersect. Graphing a System of Linear Inequalities by Hand Graph the system: x y x y 2 2 4 + ≥ − ≤ ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ Solution EXAMPLE 6 Begin by graphing the lines x y 2 + = and x y 2 4 − = using a solid line since both inequalities are nonstrict. Use the test point 0, 0 ( ) on each inequality. For example, 0, 0 ( ) does not satisfy x y 2, + ≥ so shade above the line x y 2. + = See Figure 34(a). But 0, 0 ( ) does satisfy x y 2 4, − ≤ so shade above the line x y 2 4. − = See Figure 34(b). The intersection of the shaded regions (in purple) gives the result presented in Figure 34(c). Figure 34 (0, 0) Graph of 2x 2 y # 4 x y 22 24 4 2 22 24 2 4 (b) (0, 0) Graph of x 1 y $ 2 x y 22 24 4 2 22 24 2 4 (a) x y 22 24 4 2 22 24 2 4 x 1 y $ 2 2x 2 y # 4 Graph of (c) 2x 2 y 5 4 x 1 y 5 2 Graphing a System of Linear Inequalities Using a Graphing Utility Graph the system: x y x y 2 2 4 + ≥ − ≤ ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ EXAMPLE 7
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