6.5 Systems of Inequalities

Transcription

1 6.5 Systems of Inequalities Linear Inequalities in Two Variables: A linear inequality in two variables is an inequality that can be written in the general form Ax + By < C, where A, B, and C are real numbers and A and B are not both zero. The inequality symbol can be <, >,,. Example: Each of the following is a linear inequality in two variables. a. x + 5y > 8 b. 7 x 5y + c. 8x 3 4y d. 3x 6y < 3 Example: Rewrite the linear inequality 7 x 6 y + 5in the general form Ax + By < C. Solution: In the general form Ax + By < C the x and y values are on the left side of the inequality and the constant is on the right side of the inequality. We will use the same techniques that we learned when solving linear equations in two variables. In this case all we need to do is use the addition property of inequality to subtract -6y from both sides of the equation. 7x 6y + 5 7x 6y 5 7x + 6y 5 In the last step, both sides of the inequality were multiplied by (-1). This was only done in order to reduce the number of negative signs in the inequality. Recall that when multiplying an inequality by a negative number, the direction of the inequality sign must be reversed. Example: Rewrite the linear equation < 3(7x 4) 5y + 3in the general form Ax + By < C. Solution: Again, we will use basic equation solving techniques. < 3(7x 4) 5y + 3 < 1x 1 5y + 3 < 1x 5y 9 11 < 1x 5y 1x 5y > 11 When switching the left hand and right hand expressions, don t forget to switch the inequality sign. In other words, if Ax + By < C then, C > Ax + By

2 Solutions of Linear Inequalities in Two Variables: The solution to a linear inequality in two variables is an ordered pair that when substituted for the variables will make the inequality true. Example: Show that the ordered pair (-, 1) is a solution to the linear equation 5 x y < 8. Solution: Substitute the ordered pair (-, 1) into the inequality 5 x y < 8. 5x y < 8 5( ) (1) < 8 10 < 8 1 < 8 Because this is a true statement, the ordered pair is a solution to the inequality. Example: Determine if the ordered pair (-4, 5) is a solution to the linear inequality 3x ( y 4) + 1 > x 5. Solution: Substitute the ordered pair (-4, 5) into the inequality 3x ( y 4) + 1 > x 5. 3x ( y 4) + 1 > x 5 3( 4) (5 4) + 1 > ( 4) 5 1 (1) + 1 > > > 13 Because this is NOT a true statement, the ordered pair is not a solution to the inequality. Example: Determine if the ordered pair (, 5) is a solution to the linear inequality x 3y > 1. Solution: Substitute the ordered pair (, 5) into the inequality x 3y > 1. x 3y > 1 () 3(5) > > 1 19 > 1 The ordered pair (, 5) is not a solution to the inequality.

3 The Graph of a Linear Inequality: Just as with linear equations in two variables, a linear inequality in two variables will have an infinite number of solutions. The big difference between the two is that with a linear equation the solutions are all the ordered pairs which lie on the graph of the equation (the line) but with a linear inequality the solutions will be a region of the coordinate plane which is divided by the graph of the equation (line). Example: Given the graphs of the equation y = x + and the inequality y < x + determine if the points (, 4) and (4, -4) are solutions of each by inspecting the graph. Solution: It can also be determined whether or not the points are solutions by substituting them into the equation or inequality as demonstrated below. (, 4) (4, -4) (, 4) (4, -4) y < x + y < x + y = x + y = x + 4 < + 4 < 4 4 < < 6 4 = + 4 = 4 4 = = 6 Boundary Lines: Notice in the graph of the inequality above that the points on the line itself are NOT solutions to the inequality. This will always be the case when a less than (<) or greater than (>) symbol is used. In this case, we call the equation y = x + the boundary line for the graph of the inequality y < x +. This means that the equation y = x + forms a boundary between the solution region and non-solution region of the coordinate plane but is not a part of the solution region. The boundary equation is simply the equation created from the inequality by replacing the inequality symbol with an equal sign. Boundary lines may be graphed as a dashed line or a solid line. 1. Dashed Boundary Line: Less Than (<) or Greater Than (>). Solid Boundary Line: Less Than or Equal To ( ) or Greater Than or Equal To ( )

4 Procedure for Graphing Inequalities: 1. Graph the corresponding equation. This is the boundary line. Determine if the boundary line is a solid or dashed line.. Choose a test point in either half plane and substitute the coordinates of the point into the inequality. If the resulting inequality is true, that the graph of the inequality is the half-plane containing the test point. If not true, then the other half-plane is the solution. 3. Shade the half-plane that represents the solution to the inequality. Example: Graph the inequality y > x 3. Solution: The boundary equation is y = x 3. Graph this equation using previously learned techniques to obtain the following graph. Because the inequality is a (>) symbol the boundary line will be dashed. Select a test point somewhere in either half-plane. The point (0, 0) is always an easy point to work with so I generally use it unless it lies on the boundary line. y > x 3 0 > (0) 3 0 > 3 True Because the test point (0, 0) makes the inequality true, the half-plane containing (0, 0) will be the solution region to the inequality. Remember that any point in the shaded region is a solution to the inequality.

5 Example: Graph the inequality 1 y x +. 1 Solution: The boundary equation is y = x +. Graph this equation using previously learned techniques to obtain the graph to the right. Because the inequality is a ( ) symbol the boundary line will be solid. Select a test point somewhere in either half-plane. The point (0, 0) is always an easy point to work with so I generally use it unless it lies on the boundary line. 1 y x (0) + 0 Because the test point (0, 0) makes the inequality NOT true, the half-plane NOT containing (0, 0) will be the solution region to the inequality. Remember that any point in the shaded region is a solution to the inequality. Example: Graph the inequality x 3y < 1. Solution: The boundary equation is x 3y = 1 This can be graphed using the intercepts. The boundary line will be dashed because of the (<) symbol and the test point (0, 0) is a solution to the inequality so the top half of the coordinate plane is the shaded (solution) region.

6 Graphing a Nonlinear Inequality in Two Variables: A nonlinear inequality in two variables may be graphed in much the same way as a linear inequality. Example: Graph the inequality y 4 x Solution: The boundary equation is y = 4 x which can be graphed using transformations. Using (0, 0) as a test point we obtain: y 4 x Because this is a true statement, (0, 0) is a solution and the region containing the point (0, 0) is a solution. Example: Graph the inequality x + y > 4 Solution: The boundary equation is x + y = 4 which is a circle of radius and centered at the origin.. Using (0, 0) as a test point we obtain: x + y > 4 0 > 4 > 4 Because this is a false statement, (0, 0) is NOT a solution to the inequality and the region containing the point (0, 0) is not in the solution region of the system..

7 Solving a System of Inequalities by Graphing: A solution of a system of linear inequalities in two variables is an ordered pair that makes both inequalities in the system true. Example: Graph the solution set of the system. Solution: Graph both inequalities on the coordinate plane using previously learned techniques. Select a test point (0, 0) and test the point in both equations. The point (0, 0) is a solution to the inequality y x 5. The solution region is colored yellow. The point (0, 0) is not a solution to the inequality y < 3x. The solution region is colored orange. The solution region of the system is represented by the overlapping solution regions. y x 5 y < 3x Example: Graph the solution set of the system Solution: Graph both inequalities on the coordinate plane using previously learned techniques. Select (0, 0) as a test point. The point (0, 0) is not a solution to the inequality x y. The point (0, 0) is a solution to the inequality y x 4. The solution region is the region between the curves and is colored gray. x y y x 4

8 8.5-Applications: Example: On a ski vacation Josh decided to divide lodging between large resorts and small resorts. Josh wants to spend at least 5 nights in lodging. At least one night should be spent at a large resort. Large resorts average $00 per night and small resorts average $100 per night. Josh s budget will permit no more than $700 for lodging. Write a system of inequalities that will model the above conditions. Graph the solution set. What is the greatest number of nights that Josh can stay at a large resort and still stay within his budget? Solution: Let x represent the number of nights in a large resort and y represent the number of nights at a small resort. x + y 5 x 1 00x + 100y 700 Using the point (0, 0) as a test point on all three inequalities, it can be determined that the enclosed triangular area is the solution region to the system. The greatest number of nights Josh can stay at a large resort is nights.