Explanation Method used: Graphing a Linear Inequality in Two Variables: 1. Replace the inequality symbol with an equal sign and graph the corresponding linear equation. Draw a solid line if the original inequality contains a ≥ or ≤ symbol. Draw a dashed line if the original inequality contains a > or < symbol. 2. Choose a test point from one of the half–planes. (Do not choose a point on the line.) Substitute the coordinates of the test point into inequality. 3. If a true statement results, shade the half-plane containing this test point. If a false statement results, shade the half-plane not containing this test point. Calculation: The given system of inequalities is { x − y ≤ 1 x ≥ 2 . To sketch the graph of the solution set of the above system, first sketch the graph of the each inequality. Use the above method and obtain the graph of the each inequality is as follows. The graph of the inequality x − y ≤ 1 and x ≥ 2 is as follows. Step 1: Replace the inequality symbol by = and graph the linear equation: On replacing the symbol < by = , the two equations becomes x − y = 1 and x = 2 . Use the intercepts concept to graph the line x − y = 1 . To find the x -intercept: Set y = 0 , x − 0 = 1 x = 1 Since, the x -intercept is 1, so the line passes through ( 1 , 0 ) . To find the y -intercept: Set x = 0 , 0 − y = 1 − y = 1 y = − 1 Since, the y -intercept is − 1 , thus the line passes through ( 0 , − 1 ) . Similarly, the x = 2 is a vertical line with x -intercept 2 . Thus the line passes through the point ( 2 , 0 ) . Use above intercepts and draws the graph of the line x − y = 1 as a solid line and x = 2 as a solid line as shown below in Figure 1. From Figure 1, it is observed that the both lines passes through a common point ( 2 , 1 ) . Step 2: Choose a test point from one of the half–planes. (Do not choose a point on the line.) Substitute the coordinates of the test point into inequality. The line x − y = 1 divides the x y -plane into three parts, namely the line itself and the two half-planes. The test point in one half-plane satisfy x − y ≥ 1 and the test point in other half-plane satisfy x − y ≤ 1 . To find half-plane to which the solution of x − y ≤ 1 belongs, choose a test point ( 0 , 0 ) from either of the half-planes

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Author: Blitzer

Publisher: RENT PEARS

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Chapter 5.5, Problem 39E

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To sketch: The graph of the solution set of the system of inequalities {x−y≤1x≥2 or indicate that the system has no solution.

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Chapter 5.5, Problem 38E

Chapter 5.5, Problem 40E

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The graph of the solution set of the system of inequalities { x − y ≤ 1 x ≥ 2 or indicate that the system has no solution.

Solution Summary: The author explains how to sketch the solution set of the system of inequalities by replacing the inequality symbol with an equal sign and graphing the linear equation.

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To sketch: The graph of the solution set of the system of inequalities {x−y≤1x≥2 or indicate that the system has no solution.

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