Math 20: Foundations FM20.8 Demonstrate understanding of systems of linear inequalities in two variables. E. Above or Below the Line.

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Math 20: Foundations FM20.8 Demonstrate understanding of systems of linear inequalities in two variables. E. Above or Below the Line Slide 2 Remembering Inequalities Slide 3 What DO YOU Think? p. 293 Slide 4 1. Graphing a Linear Inequality FM20.8 Demonstrate understanding of systems of linear inequalities in two variables. Slide 5 1. Graphing a Linear Inequality Slide 6 Slide 7 Slide 8 What we do is use test points to determine which side of the line we are concerned about. Select a point on one side of the line and sub it into the inequality if the inequality is solved that is the side of the line we shade Slide 9 Slide 10 The area we shade is called the Solution Region When we graph a line the Cartesian plane is cut into tow halves called Half Planes A Continuous line is a line that contains real numbers. All numbers are included so the line is solid (continuous). Slide 11 Example 1 Slide 12 Slide 13 Example 2 Slide 14 x= results in a Vertical Line y= results in a Horizontal Line Slide 15 Example 3 Slide 16 Summary p.302 Slide 17 Slide 18 Slide 19 Practice Ex. 6.1 (p.303) #1-12 #4-12 Slide 20 2. Systems of Linear Inequalities FM20.8 Demonstrate understanding of systems of linear inequalities in two variables. Slide 21 2. Systems of Linear Inequalities Slide 22 What combinations of morning and full-day students can the school accommodate and stay within the weekly snack budget? Slide 23 Summary p.307 Slide 24 Slide 25 Practice Ex. 6.2 (p.307) #1-2 Slide 26 3. Using the Graphs to Solve Problems FM20.8 Demonstrate understanding of systems of linear inequalities in two variables. Slide 27 3. Using the Graphs to Solve Problems What combinations of boats should the company make each day? Slide 28 Is every point in the region a possible solution to the problem? How would the graph change if fewer than 25 boats were made each day? All whole points with whole number coordinates in the solution region are valid, but are they all reasonable? Slide 29 Example 1 Slide 30 Example 2 Slide 31 Summary p.317 Slide 32 Slide 33 Practice Ex. 6.3 (p.317) #1-4 odds in each, 5-10 #4-12 Slide 34 4. Creating Optimization Problems FM20.8 Demonstrate understanding of systems of linear inequalities in two variables. Slide 35 4. Creating Optimization Problems Optimization Problem A problem where a quantity must be maximized or minimized following a set of guidelines or conditions. Constraint A limiting condition of the optimization problem being modeled, represented by a linear inequality. Objective Function In an optimization problem, the equation that represents the relationship between the two variables in the system of linear inequalities and the quantity to be optimized. Feasible Region The solution region for a system of linear inequalities that is modeling an optimization problem. Slide 36 Investigate the Math p.324 Slide 37 Example 1 Slide 38 Example 2 Slide 39 Summary p.329 Slide 40 Slide 41 Practice Ex. 6.4 (p.330) #1-8 #2-9 Slide 42 5. Optimization Solutions FM20.8 Demonstrate understanding of systems of linear inequalities in two variables. Slide 43 Explore the Math p.332 This is the same Problem from the beginning of last day dealing with the toy cars. So we dont have to set it up an graph it again we can use our results from last day. Slide 44 Slide 45 Summary p.333 Slide 46 Practice Ex. 6.5 (p.334) #1-3 Slide 47 6. Determine the Max and Min FM20.8 Demonstrate understanding of systems of linear inequalities in two variables. Slide 48 6. Determine the Max and Min The process that we have been developing and will finish today is called linear programming Linear Programming is used to find the solution in the feasible region result in the optimal solutions of the objective functions Slide 49 Slide 50 What happens if the optimal points (intersection points of constraints) do not land on a whole number coordinate? We solve the two inequalities like they where linear equations using the substitution or elimination strategies. This gives us the point where the two lines intersect. Slide 51 Example 1 Slide 52 Summary p.341 Slide 53 Slide 54 Practice Ex. 6.6 (p.341) #1-15 #3-15,17