The solutions to linear inequalities in two variables, such as x y 1, are ordered pairs of the form (x, y) that make the inequality true. We graph a linear inequality in two variables on a rectangular coordinate system. EXAMPLE 1 In-Class Example 1 Shown here is the graph of x y 1. Find three points that solve x y 1 and three points that are not in the solution set. x y 5 x y 1 5 5 5 Answer: in solution set (0, 2), (0, 1), (2, 4); not in solution set (0, 2), (4, 1), (2, 5) Shown here is the graph of x y 1. Find three points that solve x y 1, and fi nd three points that are not in the solution set. x y 5 x y 1 x y 1 5 5 5 Solution The solution set of x y 1 consists of all points either on the line or in the shaded region. Any point on the line or in the shaded region will make x y 1 true. Solutions of Check by Substituting x y 1 into x y 1 (1, 3) 1 3 1 ✓ (4, 1) 4 1 1 ✓ (2, 1) (on the line) 2 (1) 1 ✓ (1, 3), (4, 1), and (2, 1) are just some points that satisfy x y 1. There are infi nitely many solutions. Not in the Solution Verify by Substituting Set of x y 1 into x y 1 (0, 0) 0 0 1 False (4, 1) 4 1 1 False (2, 3) 2 (3) 1 False (0, 0), (4, 1), and (2, 3) are just three points that do not satisfy x y 1. There are infi nitely many such points. The points in the unshaded region are not in the solution set. x y 5 (1, 3) (0, 0) (4, 1) (4, 1) 5 5 5 (2, 1) (2, 3) Points in the shaded region and on the line are in the solution set. For the inequality in Example 1, notice that if you choose any ordered pair in the shaded region or on the line and substitute its x- and y-values into the inequality, you will get a true statement. If the inequality in Example 1 had been x y 1, then the line would have been drawn as a dotted line, and all points on the line would not be part of the solution set. www.mhhe.com/messersmith SECTION 4.5 Linear Inequalities in Two Variables 287
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