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**Linear Equations in One Variable**

Type Example Solution Linear equations x – 8 = 3(x + 5) A number in one variable x = -23 Graph

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**Linear Inequalities in One Variables**

Type Example Solution Linear inequalities –3x + 5 > A set of numbers; in one variable x < an interval Graph

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**Linear Equations in Two Variables**

Type Example Solution Linear equations x + y = A set of ordered in two variables pairs; a line Graph

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**An ordered pair is a solution of the linear inequality if it makes the inequality a true statement.**

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Example Determine whether (1, 5) and (6, –2) are solutions of the inequality 3x – y < 5. Solution 3(1) – –2 < 5 TRUE 3x – y < 5 3(6) – (–2) 5 20 < 5 FALSE 3x – y < 5 The pair (1, 5) is a solution of the inequality, but (6, –2) is not.

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**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 > –14 0 > –14 True, so shade the side containing (0,0).

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**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.

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**y = x Example Graph Solution First: Graph the boundary**

-3 2 -2 3 -1 1 6 5 4 y = x -4 -5 Example Graph Solution First: Graph the boundary y = 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 get We finish drawing the solution set by shading the half-plane that includes (0, 1).

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a) b) c) d)

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