SECTION 3.2: Graphing Linear Inequalities

Transcription

1 6 SECTION 3.2: Graphing Linear Inequalities GOAL: Graphing One Linear Inequality Example 1: Graphing Linear Inequalities Graph the following: a) x + y = 4 b) x + y 4 c) x + y < 4 Graphing Conventions: 1. Dotted line vs. Solid line 2. Not shaded vs. Shaded

2 7 Steps to Graphing an Inequality: 1. Graph the equality; decide what kind of line to use. a. For example, if your equation is ax + by < c, then 2. Test a point not on the line; shade the true side. Example 2: Graphing an Inequality Graph the solutions to 2x + 4y 8. Example 3: Graphing Vertical Inequalities Graph the solutions to x < 4. Graphing on a Calculator 1. Solve for y in ax + by = c and graph the result in the [Y=] screen on your calculator. 2. Test a point to determine if above or below the line is true 3. For more clarity on your calculator screen, use shading. That is, shade the side. You do this by moving the cursor all the way to the left and hitting the [ENTER] key until you get the correct shading.

3 8 Example 4: Graphing on a Calculator Graph the solutions to 6x 2y 8 on your calculator. Example 5: Graphing Vertical Inequalities Graph the solutions to x 2 on your calculator. NOTE: To trick your calculator into graphing lines of the form x = a on your calculator, use the equation: Graphing a System of Linear Inequalities Definition: Feasible Region Suppose you have a system of linear inequalities. Then the set of points satisfying of the inequalities is called the. Graphically this is the region where all of the shading (or the on the calculator). Definition: Bounded (feasible region) A feasible region is if it can be enclosed inside of a. Definition: Corner points A is a point on the corner of a.

4 9 Example 6: Graphing a Feasible Region a) Graph the solutions to the following system of linear inequalities: x + y 0 x 4 y 1 b) Is the solution set (i.e. the feasible region) bounded or unbounded? c) Find the corner points.

5 10 (Group) Example 7: Graphing a Feasible Region a) Graph the solutions to the following system of linear inequalities: x + y 4 2x + y 6 y 0 x 0 b) Is the solution set (i.e. the feasible region) bounded or unbounded? c) Find the corner points.

6 11 Example 7: Finding System of Linear Inequalities for a Region a) Find the system of linear inequalities that corresponds to the system shown below. b) Find all the corner points of the feasible region.

7 12 Example 8: Nutrition Recall Example 2 from Section 3.1: An individual needs a daily supplement of at least 500 units of vitamin C and 200 units of vitamin E and agrees to obtain this supplement by eating two foods, I and II. Each ounce of food I contains 40 units of vitamin C and 10 units of vitamin E, and each ounce of food II contains 20 units of vitamin C and 20 units of vitamin E. The total supplement of these two foods must be at most 30 ounces. Unfortunately, food I contains 30 units of cholesterol per ounce and food II contains 20 units of cholesterol per ounce. Set up the linear equations to find the appropriate amounts of the two food supplements so that cholesterol is minimized. For this problem, we found the objective function to be: cholesterol = 30x + 20y, where x was the ounces of food I consumed and food y was the ounces of food 2 consumed. We also found the problem constraints to be 40x + 20y 500, 10x + 20y 200, and x + y 30. The nonnegative constraints were x 0 and y 0. Graph the feasible region for the ounces of food I and food II this individual can eat.

8 13 Example 9: Application of a Feasible Region Recall Example 5 from Section 3.1: A baker has 23 pounds of sugar to make donuts and fritters. Donuts sell for $0.50 each and fritters sell for $1.50 each. A batch of 24 donuts requires 1.5 pounds of sugar and a batch of 15 fritters requires 2 pounds of sugar. The baker wants to have at least four times as many donuts as fritters. Write the equations to determine how many of each item the baker should make to maximize revenue. For this problem, we found the objective function to be: R x =.5d + 1.5f. We also found the problem constraints to be d 4f and!.! d +! f 23, and the nonnegative!"!" constraints to be d 0 and f 0. Graph the feasible region for the number of donuts and fritters the baker can make.