SECTION 11.7 Systems of Inequalities 831 11.7 Systems of Inequalities Now Work the ‘Are You Prepared?’ problems on page 837. • Solving Linear Inequalities (Section A.9, pp. A80–A81) • Lines ( Section 1.5 , pp. 32 – 43 ) • Circles ( Section 1.6 , pp. 48 – 52 ) • Solving Inequalities Involving Quadratic Functions (Section 3.5, pp. 179–181) • Quadratic Functions and Their Properties (Section 3.3, pp. 157–166) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Graph an Inequality by Hand (p. 831) 2 Graph an Inequality Using a Graphing Utility (p. 833) 3 Graph a System of Inequalities (p. 834) Section A.9 discusses inequalities in one variable. This section discusses inequalities in two variables. Examples of Inequalities in Two Variables (a) x y 3 6 + ≤ (b) x y 4 2 2 + < (c) y x 2 > EXAMPLE 1 1 Graph an Inequality by Hand An inequality in two variables x and y is satisfied by an ordered pair a b , ( ) if, when x is replaced by a and y by b, a true statement results.The graph of an inequality in two variables x and y consists of all points x y , ( ) whose coordinates satisfy the inequality. Graphing an Inequality by Hand Graph the linear inequality: x y 3 6 + ≤ Solution EXAMPLE 2 Begin by graphing the equation x y 3 6 + = formed by replacing (for now) the ≤ symbol with an sign. = The graph of the equation is a line. See Figure 27(a). This line is part of the graph of the inequality Figure 27 (5, 5) (21, 2) (22, 22) (4, 21) x y 26 6 5 22 (b) Graph of 3x 1 y # 6 (5, 5) (21, 2) (22, 22) (4, 21) x y 26 6 5 22 (a) 3x 1 y 5 6 (continued) ‘Are You Prepared?’ Answers 1. x y 22 22 2 2 (21,21) (0, 2) 2. x y 25 5 5 25 (22, 0) (0, 24) (2, 0) 3. x y 25 5 5 25 (21, 0) (1, 0) 4. x y (22, 0) (0, 1) (0, 21) (2, 0) 5 25 25 5
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