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1. 2. 3. 16.2 Math Pacing Graphing Inequalities in Two Variables

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Like a linear equation in two variables, the solution set of an inequality in two variables is graphed on a coordinate plane. The solution set of an inequality in two variables is the set of all ordered pairs that satisfy the inequality.

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Example 6-1a From the set {(3, 3), (0, 2), (2, 4), (1, 0)}, which ordered pairs are part of the solution set for Use a table to substitute the x and y values of each ordered pair into the inequality. false 01 true 42 false 20 true 33 True or False yx Ordered Pairs that Satisfy an Inequality

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Example 6-1a Answer: The ordered pairs {(3, 3), (2, 4)} are part of the solution set of. In the graph, notice the location of the two ordered pairs that are solutions forin relation to the line. Ordered Pairs that Satisfy an Inequality

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Example 6-1b From the set {(0, 2), (1, 3), (4, 17), (2, 1)}, which ordered pairs are part of the solution set for Answer: {(1, 3), (2, 1)} Ordered Pairs that Satisfy an Inequality

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Graphing Inequalities in Two Variables The solution set for an inequality in two variables contains many ordered pairs when the domain and range are the set of real numbers. The graphs of all these ordered pairs fill a region on the coordinate plane called a half-plane. An equation defines the boundary or edge for each half-plane.

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Consider the graph of y > 4. First determine the boundary by graphing y = 4, the equation obtained by replacing the inequality sign with an equals sign. Since the inequality involves y- values greater than 4, but not equal to 4, the line should be dashed. The boundary divides the coordinate plane into two half-planes.

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To determine which half-plane contains the solution, choose a point from each half-plane and test it in the inequality. The half-plane containing (5, 6), the point that satisfies the inequality, contains the solution. Shade that half-plane. Try (3, 0). y > 4(y = 0) 0 > 4FALSE Try (5, 6). y > 4(y = 6) 6 > 4TRUE graph of y > 4.

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Example 6-2a Step 1 Solve for y in terms of x. Original inequality Add 4x to each side. Simplify. Divide each side by 2. Simplify. Graph an Inequality

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Example 6-2a Step 2 Graph Sincedoes not include values when the boundary is not included in the solution set. The boundary should be drawn as a dashed line. Step 3 Select a point in one of the half-planes and test it. Let’s use (0, 0). Original inequality false y = 2x + 3 Graph an Inequality

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Example 6-2a Answer:Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane. y = 2x + 3 Graph an Inequality

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Example 6-2a Answer:Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane. CheckTest the point in the other half-plane, for example, (–3, 1). Original inequality Since the statement is true, the half-plane containing (–3, 1) should be shaded. The graph of the solution is correct. y = 2x + 3 Graph an Inequality

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Example 6-2b Answer: Graph an Inequality

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