2 Linear Equations in One Variable Type Example SolutionLinear equations x – 8 = 3(x + 5) A numberin one variable x = -23Graph
3 Linear Inequalities in One Variables Type Example SolutionLinear inequalities –3x + 5 > A set of numbers;in one variable x < an intervalGraph
4 Linear Equations in Two Variables Type Example SolutionLinear equations x + y = A set of orderedin two variables pairs; a lineGraph
5 An ordered pair is a solution of the linear inequality if it makes the inequality a true statement.
6 ExampleDetermine whether (1, 5) and (6, –2) are solutions of the inequality 3x – y < 5.Solution3(1) ––2 < 5TRUE3x – y < 53(6) – (–2) 520 < 5FALSE3x – y < 5The pair (1, 5) is a solution of the inequality, but (6, –2) is not.
7 Example: Graph 7x + y > –14 y First : Graph the boundary line 7x + y = –14 or y = -7x - 14(0, 0)Second: Pick a test point not on the boundary: (0,0)Test it in the original inequality.7(0) + 0 > –140 > –14True, so shade the side containing (0,0).
8 To Graph a Linear Inequality in Two Variables 1. Graph the boundary line found by replacing the inequality sign with an equal sign. If the inequality sign is > or <, graph a dashed boundary line (indicating that the points on the line are not solutions of the inequality). If the inequality sign is or , graph a solid boundary line (indicating that the points on the line are solutions of the inequality).2. Choose a point, not on the boundary line, as a test point. Substitute into the original inequality. (0,0) or (1,0) or (0,1)3. If a true statement is obtained in Step 2, shade the half-plane that contains the test point. If a false statement is obtained, shade the half-plane that does not contain the test point.
9 y = x Example Graph Solution First: Graph the boundary -32-23-11654y = x-4-5ExampleGraphSolutionFirst: Graph the boundaryy = x ( Solid boundary line, > )(0, 1)Second: We choose a test point on one side of the boundary, say (0, 1).Substituting into the inequality we getWe finish drawing the solution set by shading the half-plane that includes (0, 1).