Chapter 3: Section 3-2 Graphing Linear Inequalities

Transcription

1 Chapter : Section - Graphing Linear Inequalities D. S. Malik Creighton Universit, Omaha, NE D. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities / 9

2 Geometric Approach to Solve Linear Programming Problems Graph the constraints of the problem Find the solutions set (feasible region) Find the corner points of the solution set Use corner points to find a solution, if it eists D. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities / 9

3 Graphing Linear Inequalities Eample Consider the inequalit: +. The first step in drawing the graph of an inequalit is to change the inequalit into an equalit and then draw the graph of the equalit. So first we draw the graph of the equalit + =. Two points on this line are: (, ) 0 (0, ) 0 (, 0). S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities / 9

4 Eample (Two points on the line are: (0, ) and (, 0)) The graph of + = is: + = 0 Select a point, which is not on the line. For eample, (0, ) is such a point.. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 4 / 9

5 Eample Determine, whether (0, ) satisfies the given inequalit, which is +. Now 0 + = 6, which is true. So the point (0, ) is a solution of the given inequalit. Shade the portion of the graph containing the point (0, ) to draw the graph of +. The graph of + is: + = + > 0. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 5 / 9

6 Eample In this eample, we draw the graph of <. First we draw the graph of =. Two points on this line are (, 0) and (0, ). The graph of the line is shown in (a). (Note that the inequalit is <, so we draw a dotted line.) Net we choose a point which is not on the line. (0, 0) is one such point. We can use (0, 0) as a test point. Now 0 0 = 0 <, which is false. So we shade the region that does not contain the point (0, 0) and obtain the graph in (b). 0 < 0 (a) (b). S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 6 / 9

7 Eample Consider the following inequalities: + 6 and.5 The graphs of these inequalities are: + > > We determine the region in the - plane that simultaneousl satisfies these inequalities.. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 7 / 9

8 Eample The graphs of the inequalities: + 6 and.5 We draw the graph of the these inequalities in the same - plane, see Figure (a). Two find the region that simultaneousl satisfies these inequalities, we choose the commonl shaded region as shown in Figure (b). + > 6 + > 6.5 > 0.5 > 0. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 8 / 9

9 Eample Consider the following inequalities: +, 5, 6. The region that simultaneousl satisfies these equations is shown in the following figure: + = A B C 0 = 5 6 = The corner points of the shaded region are A, B, and C.. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 9 / 9

10 Eample Point A is the intersection of the lines + = and = 5. Point B is the intersection of the lines 6 = and = 5. Point C is the intersection of the lines + = and 6 =. To find the coordinates of the point A, solve the equations + = and = 5 for and to get ( 9 8, ) 8. To find the coordinates of the point B, solve the equations 6 = and = 5 for and to get ( 0, 8 ) 5. To find the coordinates of the point C, solve the equations + = and 6 = for and to get ( 9, 5 ) 9.. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 0 / 9

11 Definition (Bounded and Unbounded Sets) A set of points in the - plane is said to be bounded if it can be contained in a circle with center (0, 0). Otherwise, the set is said to be unbounded. Definition (Solutions set) The solution set of a sstem of linear inequalities is the set of points that simultaneousl satisf all of the inequalities. Remark When we graph a sstem of inequalities, then the solution set is the set of all points in the shaded region. In the graph, the shaded region is called the solution set or the feasible region. Definition (Corner points) Let S be a solution set for a sstem of linear inequalities. A point C in the - plane is called a corner point of S if ever line segment in S that contains C has C as one of its endpoints.. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities / 9

12 Eample Consider the following sstem of inequalities: + 5, 4 0, The graph of these inequalities is: = 0 B = C A = 5. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities / 9

13 Eample Point A is the intersection of the lines 4 = 0 and =. Point B is the intersection of the lines 4 = 0 and + = 5. Point C is the intersection of the lines + = 5 and =. The coordinates of A are (, ). The coordinates of B are (, 4). The coordinates of C are (, ).. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities / 9

14 Eample Consider the following sstem of inequalities: 0, 0, + 4, The graph of these inequalities is: 5 A 4 + = 4 B C = 0 The solution set is unbounded with three corner points A, B, and C.. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 4 / 9

15 Eample A is the intersection of the lines = 0 and 5 + = 0. B is the intersection of the lines 5 + = 0 and + = 4. C is the intersection of the lines = 0 and + = 4. The coordinates of A are (0, 5), and the coordinates of C are (4, 0). The coordinates of B are (, 5 ) 4. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 5 / 9

16 Eercise: Consider the following eercise described earlier. A pet store specializes in cats and bunnies. Each cat costs $9 and each bunn costs $6. The profit on each cat is $ and on each bunn is $9. The store cannot house more than 0 animals and cannot spend more than $6 to bu the pets. Under a special agreement the pet store must house at least cats. How man pets of each tpe should be housed to maimize the profit? Let be the number of cats and be the numbers of bunnies. The linear programming problem describing this problem is: The linear programming problem is: maimize: + 9 subject to:, Let us graph the constraints and determine the feasible region. D. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 6 / 9

17 The graph of the inequalities, 0, + 0, + 7 is: 40 + = = B(, 8) C(, 8) A(, 0) D(4, 0) = 7 The corner points are A(, 0), B(, 8), C (, 8), D(4, 0). D. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 7 / 9

18 Eercise Find the inequalit that represents the following graph: 0 Solution: The line passes through (, 0) and (0, ). The slope of this line is m =. The equation of the line with slope 0 0 ( ) = and -intercept (0, ) is = +, i.e., = + 6, or = 6. This implies that the required inequalit is 6 or 6. D. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 8 / 9

19 0 This implies that the required inequalit is 6 or 6. Net choose a point which is either in the shaded region or in the unshaded region. The point (0, 0) is in the unshaded region. Net evaluate the epression at (0, 0). Thus, = 0 0 = 0. If the inequalit is 6, then (0, 0) will satisf the inequalit because at (0, 0), = 0 6 is true. However, (0, 0) is not in the shaded region, so it should not satisf the inequalit. Hence, the inequalit that represents the above graph is 6. D. S. Malik Creighton Universit, Omaha, NE Chapter () : Section - Graphing Linear Inequalities 9 / 9