Section 7.8 from Basic Mathematics Review by Oka Kurniawan was developed by OpenStax College, licensed by Rice University, and is available on the

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1 Section 7.8 from Basic Mathematics Review by Oka Kurniawan was developed by OpenStax College, licensed by Rice University, and is available on the Connexions website. It is used under a Creative Commons Attribution 3.0 Unported license.

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3 508 CHAPTER 7. GRAPHING LINEAR EQUATIONS AND INEQUALITIES IN ONE AND TWO VARIABLES Solution to an Inequality in Two Variables Recall that when working with linear equations in two variables, we observed that ordered pairs that produced true statements when substituted into an equation were called solutions to that equation. We can make a similar statement for inequalities in two variables. We say that an inequality in two variables has a solution when a pair of values has been found such that when these values are substituted into the inequality a true statement results. The Location of Solutions in the Plane As with equations, solutions to linear inequalities have particular locations in the plane. All solutions to a linear inequality in two variables are located in one and only in one entire half-plane. For example, consider the inequality 2x + 3y 6 All the solutions to the inequality2x + 3y 6 lie in the shaded half-plane. Example 7.51 Point A (1, 1)is a solution since 2x + 3y 6 2 (1) + 3 ( 1) 6? 2 3 6? 1 6. True Example 7.52 Point B (2, 5)is not a solution since 2x + 3y 6 2 (2) + 3 (5) 6? ? False Method of Graphing The method of graphing linear inequalities in two variables is as follows: 1. Graph the boundary line (consider the inequality as an equation, that is, replace the inequality sign with an equal sign). a. If the inequality is or, draw the boundary line solid. This means that points on the line are solutions and are part of the graph.

4 509 b. If the inequality is < or >, draw the boundary line dotted. This means that points on the line are not solutions and are not part of the graph. 2. Determine which half-plane to shade by choosing a test point. a. If, when substituted, the test point yields a true statement, shade the half-plane containing it. b. If, when substituted, the test point yields a false statement, shade the half-plane on the opposite side of the boundary line Sample Set A Example 7.53 Graph 3x 2y Graph the boundary line. The inequality is so we'll draw the line solid. Consider the inequality as an equation. 3x 2y = 4 (7.1) x y (x, y) (0, 2) ( 4 3, 0) Table Choose a test point. The easiest one is(0, 0). Substitute (0, 0) into the original inequality. 3x 2y 4 3 (0) 2 (0) 4? 0 0 4? 0 4. True (7.2)

5 510 CHAPTER 7. GRAPHING LINEAR EQUATIONS AND INEQUALITIES IN ONE AND TWO VARIABLES Shade the half-plane containing (0, 0). Example 7.54 Graph x + y 3 < Graph the boundary line: x + y 3 = 0. The inequality is < so we'll draw the line dotted. 2. Choose a test point, say (0, 0). Shade the half-plane containing (0, 0). x + y 3 < < 0? 3 < 0. True (7.3)

6 511 Example 7.55 Graph y 2x. 1. Graph the boundary line y = 2x. The inequality is, so we'll draw the line solid. 2. Choose a test point, say (0, 0). y 2x 0 2 (0)? 0 0. True Shade the half-plane containing (0, 0). We can't! (0, 0) is right on the line! Pick another test point, say (1, 6). y 2x 6 2 (1)? 6 2. False Shade the half-plane on the opposite side of the boundary line. Example 7.56 Graph y > Graph the boundary line y = 2. The inequality is > so we'll draw the line dotted.

7 512 CHAPTER 7. GRAPHING LINEAR EQUATIONS AND INEQUALITIES IN ONE AND TWO VARIABLES 2. We don't really need a test point. Where is y > 2?Above the line y = 2! Any point above the line clearly has a y-coordinate greater than Practice Set A Solve the following inequalities by graphing. Exercise (Solution on p. 553.) 3x + 2y 4 Exercise (Solution on p. 553.) x 4y < 4

8 513 Exercise (Solution on p. 554.) 3x + y > 0 Exercise (Solution on p. 554.) x 1

9 Exercises CHAPTER 7. GRAPHING LINEAR EQUATIONS AND INEQUALITIES IN ONE AND TWO VARIABLES Solve the inequalities by graphing. Exercise (Solution on p. 554.) y < x + 1 Exercise x + y 1 Exercise (Solution on p. 555.) x + 2y Exercise x + 5y 10 < 0

10 515 Exercise (Solution on p. 555.) 3x + 4y > 12 Exercise x + 5y 15 0 Exercise (Solution on p. 555.) y 4

11 516 CHAPTER 7. GRAPHING LINEAR EQUATIONS AND INEQUALITIES IN ONE AND TWO VARIABLES Exercise x 2 Exercise (Solution on p. 556.) x 0 Exercise x y < 0

12 517 Exercise (Solution on p. 556.) x + 3y 0 Exercise x + 4y > Exercises for Review Exercise (Solution on p. 556.) (Section 7.2) Graph the inequality 3x

13 518 CHAPTER 7. GRAPHING LINEAR EQUATIONS AND INEQUALITIES IN ONE AND TWO VARIABLES Exercise (Section 7.2) Supply the missing word. The geometric representation (picture) of the solutions to an equation is called the of the equation. Exercise (Solution on p. 556.) (Section 7.5) Supply the denominator:m = y2 y1?. Exercise (Section 7.6) Graph the equation y = 3x + 2. Exercise (Solution on p. 556.) (Section 7.7) Write the equation of the line that has slope 4 and passes through the point ( 1, 2).