In order to draw the graph of the inequality 1 we

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College Algebra
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Chapter 2 / Exercise 96
College Algebra
Gustafson/Hughes
Expert Verified
In order to draw the graph of the inequality (1), we take one point say (0, 0), in half plane I and check whether values of x and y satisfy the inequality or not. 2015-16
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College Algebra
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Chapter 2 / Exercise 96
College Algebra
Gustafson/Hughes
Expert Verified
126 MATHEMATICS We observe that x = 0, y = 0 satisfy the inequality. Thus, we say that the half plane I is the graph (Fig 6.7) of the inequality. Since the points on the line also satisfy the inequality (1) above, the line is also a part of the graph. Thus, the graph of the given inequality is half plane I including the line itself. Clearly half plane II is not the part of the graph. Hence, solutions of inequality (1) will consist of all the points of its graph (half plane I including the line). We shall now consider some examples to explain the above procedure for solving a linear inequality involving two variables. Example 9 Solve 3 x + 2 y > 6 graphically. Solution Graph of 3 x + 2 y = 6 is given as dotted line in the Fig 6.8. This line divides the xy -plane in two half planes I and II. We select a point (not on the line), say (0, 0), which lies in one of the half planes (Fig 6.8) and determine if this point satisfies the given inequality, we note that 3 (0) + 2 (0) > 6 or 0 > 6 , which is false. Hence, half plane I is not the solution region of the given inequality. Clearly, any point on the line does not satisfy the given strict inequality. In other words, the shaded half plane II excluding the points on the line is the solution region of the inequality. Example 10 Solve 3 x – 6 0 graphically in two dimensional plane. Solution Graph of 3 x – 6 = 0 is given in the Fig 6.9. We select a point, say (0, 0) and substituting it in given inequality, we see that: 3 (0) – 6 0 or – 6 0 which is false. Thus, the solution region is the shaded region on the right hand side of the line x = 2. Fig 6.7 Fig 6.8 Fig 6.9 2015-16
LINEAR INEQUALITIES 127 Fig 6.10 Example 11 Solve y < 2 graphically.
Fig 6.11 EXERCISE 6.2 Solve the following inequalities graphically in two-dimensional plane: 1 . x + y < 5 2. 2 x + y 6 3. 3 x + 4 y 4 . y + 8 2 x 5. x y 2 6. 2 x – 3 y > 6 7. – 3 x + 2 y – 6 8. 3 y – 5 x < 30 9. y < – 2 10. x > – 3. 6.5 Solution of System of Linear Inequalities in Two Variables In previous Section, you have learnt how to solve linear inequality in one or two variables graphically. We will now illustrate the method for solving a system of linear inequalities in two variables graphically through some examples. Example 12 Solve the following system of linear inequalities graphically. x + y 5 ... (1) x y 3 ... (2) Solution The graph of linear equation x + y = 5 is drawn in Fig 6.11. We note that solution of inequality (1) is represented by the shaded region above the line x + y = 5, including the points on the line. On the same set of axes, we draw the graph of the equation x y = 3 as shown in Fig 6.11. Then we note that inequality (2) represents the shaded region above 2015-16 12
128 MATHEMATICS the line x y = 3, including the points on the line.

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