SECTION 11.7 Systems of Inequalities 833 Graphing Linear Inequalities by Hand Graph: (a) y 2 < (b) y x2 ≥ Solution EXAMPLE 4 (a) Points on the horizontal line y 2 = are not part of the graph of the inequality, so the graph is shown as a dashed line. Since 0, 0 ( ) satisfies the inequality, the graph consists of the half-plane below the line y 2. = See Figure 30. (b) Points on the line y x2 = are part of the graph of the inequality, so the graph is shown as a solid line. Use 3, 0 ( ) as a test point. It does not satisfy the inequality 0 2 3. [ ] < ⋅ Points in the half-plane opposite the side containing 3, 0 ( ) satisfy the inequality. See Figure 31. Figure 30 < y 2 y y 5 2 21 23 3 5 1 21 23 25 (0, 0) 3 Graph of y , 2 5 Figure 31 ≥ y x2 x y 22 4 2 22 24 2 4 y 5 2x (3, 0) Graph of y $ 2x Now Work PROBLEM 13 2 Graph an Inequality Using a Graphing Utility Graphing utilities can also be used to graph inequalities. The steps to follow using most graphing utilities are given next. Consult your user’s manual for the steps using your graphing utility. Steps for Graphing an Inequality Using a Graphing Utility Step 1 Replace the inequality symbol by an equal sign and graph the resulting equation. This graph separates the xy -plane into two or more regions. Step 2 Select a test point P in each region. • Use a graphing utility to determine if the test point P satisfies the inequality. If the test point satisfies the inequality, then so do all the points in this region. Indicate this by using the graphing utility to shade the region. • If the coordinates of P do not satisfy the inequality, then none of the points in that region does. Graphing an Inequality Using a Graphing Utility Use a graphing utility to graph x y 3 6. + ≤ Solution EXAMPLE 5 Step 1 Begin by graphing the equation x y Y x 3 6 3 6 . 1 ( ) + = = − + See Figure 32 on the next page. Step 2 Select a test point in one of the regions and determine whether it satisfies the inequality. To test the point 1, 2 , ( ) − for example, enter 3 1 2 6. ( ) − + ≤ See Figure 33(a) on the next page. The 1 that appears indicates that the statement entered (the inequality) is true. When the point 5, 5 ( ) is tested, a 0 appears, indicating that the statement entered is false. So 1, 2 ( ) − is part of the graph of the inequality and 5, 5 ( ) is not. Shade the region containing the (continued)
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