Read the explanations, follow the examples, take notes, and complete the You Trys. In Chapter 3, we learned how to solve linear inequalities in one variable such as 2x 3 5. We will begin this section by learning how to graph the solution set of linear inequalities in two variables. Then we will learn how to graph the solution set of systems of linear inequalities in two variables. 1 Graph a Linear Inequality in Two Variables Definition A linear equality in two variables is an inequality that can be written in the form Ax By C or Ax By C where A, B, and C are real numbers and where A and B are not both zero. ( and may be substituted for and .) Here are some examples of linear inequalities in two variables. x y 3, y 1 4 x 3, x 2, y 4 Note We can call x 2 a linear inequality in two variables because we can write it as x 0y 2. Likewise, we can write y 4 as 0x y 4. The solutions to linear inequalities in two variables, such as x y 3, 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. x y 5 x y 5 5 5 The points (5, 2), (1, 4), and (3, 0) are some of the points that satisfy x y 3. There are infi nitely many solutions. The points (0, 0), (4, 1), and (2, 3) are three of the points that do not satisfy x y 3. There are infi nitely points that are not solutions. x 5 x 3 (5, 2) (1, 4) (0, 0) (3, 0) (4, 1) 5 5 5 (2, 3) Points in the shaded region and on the line are in the solution set. The points in the unshaded region are not in the solution set. SECTION www.mhhe.com/messersmith 4.4 Linear and Compound Linear Inequalities in Two Variables 187
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