832 CHAPTER 11 Systems of Equations and Inequalities Steps for Graphing an Inequality by Hand Step 1 Replace the inequality symbol by an equal sign, and graph the resulting equation. If the inequality is strict, use dashes; if it is nonstrict, use a solid mark. This graph separates the xy -plane into two or more regions. Step 2 In each region, select a test point P. • If the coordinates of P satisfy the inequality, so do all the points in that region. Indicate this by shading the region. • If the coordinates of P do not satisfy the inequality, no point in that region satisfies the inequality. RECALL The strict inequalities are < and > . The nonstrict inequalities are ≤ and ≥ . j Figure 28 + ≤ x y 4 2 2 x y –3 3 3 –3 (4, 0) (0, 0) x2 1 y2 5 4 because the inequality is nonstrict, so the line is drawn as a solid line. (Do you see why? We are seeking points for which x y 3 + is less than or equal to 6.) Now test a few randomly selected points to see whether they belong to the graph of the inequality. x y 3 6 + ≤ Conclusion 4, 1 ( ) − 3 4 1 11 6 ( ) ⋅ + − = > Does not belong to the graph 5, 5 ( ) 3 5 5 20 6 ⋅ + = > Does not belong to the graph 1, 2 ( ) − 3 1 2 1 6 ( ) − + = − ≤ Belongs to the graph 2, 2 ( ) − − 3 2 2 8 6 ( ) ( ) − + − = − ≤ Belongs to the graph Look again at Figure 27(a) on the previous page. Notice that the two points that belong to the graph both lie on the same side of the line, and the two points that do not belong to the graph lie on the opposite side.As it turns out, all the points that satisfy the inequality will lie on one side of the line or on the line itself. All the points that do not satisfy the inequality will lie on the other side. The graph of x y 3 6 + ≤ consists of all points that lie on the line or on the same side of the line as 1, 2 ( ) − and 2, 2. ( ) − − This graph is shown as the shaded region in Figure 27(b) on the previous page. Now Work PROBLEM 15 The graph of any inequality in two variables may be obtained similarly.The steps to follow are given next. Figure 29 Ax 1 By 5 C x y Graphing an Inequality by Hand Graph: x y 4 2 2 + ≤ Solution EXAMPLE 3 Step 1 Graph the equation x y 4, 2 2 + = a circle of radius 2, with center at the origin. A solid circle is used because the inequality is not strict. Step 2 Use two test points, one inside the circle, the other outside. x y x y Inside 0, 0 : 0 0 0 4 Outside 4, 0 : 4 0 16 4 2 2 2 2 2 2 2 2 ( ) ( ) + = + = ≤ + = + = > All the points inside and on the circle satisfy the inequality. See Figure 28. Belongs to the graph Does not belong to the graph Now Work PROBLEM 17 Linear Inequalities A linear inequality is an inequality in one of the forms AxByC AxByC AxByC AxByC + < + > + ≤ + ≥ where A and B are not both zero. The graph of the corresponding equation of a linear inequality is a line that separates the xy -plane into two regions called half-planes . See Figure 29. As shown, Ax By C + = is the equation of the boundary line, and it divides the plane into two half-planes: one for which Ax By C + < and the other for which Ax By C. + > Because of this, for linear inequalities, only one test point is required.
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