6.8 Linear Inequalities in Two Variables and Systems of Linear Inequalities 367 Graphing Linear Inequalities with 2 Variables GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES 1. Mentally substitute the equal sign for the inequality sign and plot points as if you were graphing the equation. 2. If the inequality is < or >, draw a dashed line through the points. If the inequality is ≤ or ≥, draw a solid line through the points. 3. Select a test point not on the line and substitute the x- and y-coordinates into the inequality. If the substitution results in a true statement, shade in the area on the same side of the line as the test point. If the substitution results in a false statement, shade in the area on the opposite side of the line as the test point. PROCEDURE 3 4 21 22 23 24 25 y x 22 23 24 25 1 2 3 5 4 1 21 (0, 2) (2, 1) (4, 0) x 1 2y 5 4 2 5 Figure 6.30 Example 1 Graphing an Inequality Graph the inequality x y2 4. + < Solution To obtain the solution set, start by graphing x y2 4. + = Since the original inequality is strictly “less than,” draw a dashed line (Fig. 6.30). The dashed line indicates that the points on the line are not part of the solution set. The line x y2 4 + = divides the plane into three parts, the line itself and two half-planes. The line is the boundary between the two half-planes. The points in only one of the two half-planes will satisfy the inequality x y2 4. + < The points in the other half-plane will satisfy the inequality x y2 4. + > To determine the solution set of the inequality x y2 4, + < pick any point that is not on the line. The simplest point to work with is the origin, (0, 0). Substitute x 0 = and y 0 = into x y2 4. + < + < + < + < < x y2 4 Is 0 2(0) 4? 0 0 4 0 4 True Since 0 4 < is a true statement, (0, 0) is a solution and every point in the half plane containing (0, 0) also represents a solution. We indicate the solution set by shading the half-plane that contains (0, 0). The solution set is shown in Fig. 6.31. 7 Now try Exercise 17 3 4 21 22 23 24 25 y x 22 23 2524 1 2 3 5 4 2 1 21 (0, 2) (2, 1) (4, 0) x 1 2y , 4 5 Figure 6.31 Example 2 Graphing an Inequality Graph the inequality y x 2 3 2. ≥ + Solution First draw the graph of the equation y x 2. 2 3 = + Use a solid line because the inequality is greater than or equal to and the points on the boundary line are included in the solution set (see Fig. 6.32). Now pick a point that is not on the line. Take (0, 0) as the test point. ≥ + ≥ + ≥ y x 2 3 2 Is 0 2 3 (0) 2? 0 2 False Since 0 2 ≥ is a false statement, (0, 0) is not a solution. Therefore, the solution set is represented by the points on the line and the points in the half plane that does not contain the point (0, 0). The solution set is shown in Fig. 6.32. 4 5 3 2 3 1 5 4 – 1 1 –5 –3 –2 –1 –2 –3 –4 –5 y x (–3, 2) (0, 2) (–3, 0) y $ x 1 2 2 3 Figure 6.32
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