3.3 Systems of Linear Inequalities-graphing

-linear programming

SolutionThe solution to a system of inequalities is the area where the shaded regions overlap. Any point in this area will work for BOTH of the inequalities.

To Graph a Linear Inequality:ToTo Graph a Linear Inequality:

1) Solve the inequality for y *Remember: if dividing or multiplying by a negative to switch the sign

2) Graph the line * Plot the y-intercept then use the slope to find the next point * Dotted line for < and > * Solid line for < and >

3) Pick a point not on the line

4) Plug the point into the inequality*True = shade the side with the point*False = shade the other side of the line

Graph a Linear Inequality:

3) Solve the inequality for y *Remember: if dividing or multiplying by a negative to switch the sign

4) Graph the line * Plot the y-intercept then use the slope to find the next point * Dotted line for < and > * Solid line for < and >

3) Pick a point not on the line

4) Plug the point into the inequality*True = shade the side with the point*False = shade the other side of the line

5) Solve the inequality for y *Remember: if dividing or multiplying by a negative to switch the sign

6) Graph the line * Plot the y-intercept then use the slope to find the next point * Dotted line for < and > * Solid line for < and >

3) Pick a point not on the line

4) Plug the point into the inequality*True = shade the side with the point*False = shade the other side of the line

Graph the system

y < -2x + 4x > -3

Graph the system

y < 3x – 6 y > -4x + 2

Graph the system

-2y < 4x + 2 y > x + 2 y < 4

Linear Programming

-Used to find the maximum or minimum value of an objective function.

-The maximum or minimum must occur at a vertex of the feasible region.

-Find the feasible region by graphing all constraints. The feasible region is found where all of the shading overlaps.

-Determine the coordinates of the vertices of the feasible region and plug these into the objective function to find the minimum or maximum value.-Unbounded regions may have no maximum or minimum.

Find the maximum and minimum value of z = 3x + 4y subject to the following constraints:

So the corner points are (2, 6), (6, 4), and (–1, –3).

Linear Programming Problem

The available parking area of a parking lot is 600 square meters. A car requires 6 sq. meters of space

and a bus requires 30 sq. meters of space. The attendant can handle no more that 60 vehicles. If a

car is charged $2.50 to park and a bus is charged $7.50, how many of each should the attendant accept

to maximize profit?

Linear Programming

Jeff makes 2 types of wood clocks to sell at local stores. It takes him 2 hours to assemble a pine clock, which requires 1 oz. of varnish. It takes 2

hours to assemble an oak clock which takes 4 oz. of varnish. Jeff has 16 oz. of varnish and he can

work 20 hours. He makes a $3 profit on pine clocks and a $4 profit on oak clocks. How many of

each type should he make to maximize profit?

Linda makes pumpkin bread and banana bread to sell at a bazaar. A loaf of pumpkin bread requires 2 cups of flour and 2 eggs. A loaf of banana bread takes 3 cups of flour and 1 egg. Linda has 12 cups of flour and 8 eggs. She makes a $2 profit per loaf of pumpkin bread and $2 profit per loaf of banana bread. How many loaves of each should she make to maximize profit?

A biologist needs at least 40 fish for her experiment. She cannot use more than 25 perch or more than 30 bass. Each perch costs $5 and each bass costs $3. How many of each type of fish should she use in order to minimize her cost?

-Identify your variables-organize your information (charts help!)-write your objective function-write all constraints-graph all constraints-determine the vertices of the feasible region-plug the coordinates of the feasible region into the objective function-determine the maximum or minimum value-state your answer

Step by step!

Linear programming!

• Your factory makes fruit filled breakfast bars and granola bars. For each case of breakfast bars you make $40 profit. For each case of granola bars you make $55 profit. You have a maximum of 150 machine hours and it takes 6 machine hours for each case of granola bars and 2 machine hours for each case of breakfast bars. You have only 155 hours of labor available and it takes 5 hours to make one case of breakfast bars and 4 hours for a case of granola bars. How many cases of each should you produce to maximize profit?