4.02 - Systems of Inequalities | High School

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High School
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High School Physics
Academic year:

2024
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16
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WHAT IS IT? A systems of inequalities is a list of constraints that all affect the same situation. For example: A thriller ride at wonderland A rider must be at least 16 to enter the ride A rider must be at least 5’5” to enter the ride 2 constraints on one situation Can a 30 year old that is 5’1” go on the ride? Can a 6 foot tall 10 year old go on the ride? STEPS: 1. Re-write inequalities in Function form (y=ax+b) 2. Graph each inequality, forming boundary lines of polygon 3. Label each boundary with rule (y=ax+b) 4. Identify side to shade of each boundary (little arrows or marks) 5. Shade overlapped region SOLVE THE SYSTEM 3 𝑦≥− 𝑥+6 2 3𝑥 + 2𝑦 ≥ 12 1 𝑦< 𝑥 2 3 𝑦 ≥− 𝑥+6 2 11 < 𝑥 𝑦𝑦< 22 𝑃𝑂𝐼 3 1 − 𝑥+6= 𝑥 2 2 6 = 2𝑥 3=𝑥 3 1 𝑦 = − (3) + 6 𝑦= 3 2 2 3 3 = = 2 2 3, 3 2 WHAT IT IS A polygon of constraints is the region of solution of a system of linear inequalities with extremities labelled. SOLVE FOR THE SOLUTION SET 𝑥 ≥ 10 𝑦 ≥ 30 𝑥 + 𝑦 ≤ 100 𝑥≥0 𝑦≥0 𝑥 ≥ 10 GRAPH 𝑥≥0 𝑥 ≥ 10 𝑦 ≥ 30 𝑥 + 𝑦 ≤ 100 𝑥≥0 𝑦≥0 𝑦 ≥ 30 𝑦≥0 𝑥 + 𝑦 ≤ 100 𝑥 ≥ 10 GRAPH A/ 𝑦 = 30 𝑥 = 10 𝐴(10,30) B/ 𝑦 = −𝑥 + 100 𝑥 = 10 𝑦 = −10 + 100 𝑦 = 90 𝐵(10,90) C/ 𝑦 = 30 𝑦 = −𝑥 + 100 30 = −𝑥 + 100 𝑥 = 70 𝐶(70,30) B A C 𝑦 ≥ 30 𝑦≥0 𝑥≥0 𝑥 + 𝑦 ≤ 100 𝑥 ≥ 10 𝑥 ≥ 10 𝑦 ≥ 30 𝑥 + 𝑦 ≤ 100 𝑥≥0 𝑦≥0 𝐵(10,90) B 𝐶(70,30) A C 𝑦 ≥ 30 𝐴(10,30) 𝑦≥0 𝑥≥0 𝑥 + 𝑦 ≤ 100 CHECKLIST FOR POLYGON OF CONSTRAINTS (1)Clearly Identify your variables • 𝑥= 𝑦= (2)Highlight any key MATH words (3)Translate the constraints into inequalities (4)Graph the inequalities • Shade the SOLUTION SET (5)Label the corners of your shaded region (6)Find the coordinate points of the vertices QUESTION WITH CONTEXT Stanley is preparing for his big Super bowl party that he hosts every year. He always provides guests with souvenirs from the event. This year he has chosen foam fingers and hats. He allows each guest to choose one or the other (not both) Stanley has invited 300 people and knows more than 120 people will attend. He knows that at least 20 of his guests will choose the foam finger. Even though the hats cost more, he intends to have at least twice as many hats available to give away as foam fingers, but no more than 200 hats. Solve the Polygon of Constraints 𝑥 represents the number of foam fingers and 𝑦 the number of hats Stanley supplies 𝑥 ≥ 20 𝑦 ≥ 2𝑥 GRAPH 𝑥≥0 𝑥 + 𝑦 ≤ 300 𝑦 ≤ 200 𝑥 + 𝑦 > 120 𝑥 ≥ 20 𝑦 ≤ 200 # of Hats 𝑦 ≥ 2𝑥 𝑥 + 𝑦 ≤ 300 𝑥≥0 𝑦≥0 𝑦≥0 𝑥 + 𝑦 > 120 # of Foam Fingers 𝑥 ≥ 20 𝑦 ≥ 2𝑥 GRAPH 𝑥≥0 B # of Hats A/ 𝑦 = −𝑥 + 120 𝑥 = 20 𝑦 = −20 + 120 𝑦 = 100 𝐴(20,100) B/ 𝑦 = 200 𝑥 = 20 𝐵(20,200) C/ 𝑦 = 200 𝑦 = 2𝑥 200 = 2𝑥 𝑥 = 100 𝐶(100,200) D/ 𝑦 = −𝑥 + 120 𝑦 = 2𝑥 2𝑥 = −𝑥 + 120 𝑥 = 40 𝑦 = 2 40 = 80 𝐷(40,80) C 𝑦 ≤ 200 A D 𝑥 + 𝑦 ≤ 300 𝑦≥0 𝑥 + 𝑦 ≥ 120 # of Foam Fingers 𝑥 ≥ 20 𝑦 ≥ 2𝑥 GRAPH 𝑥≥0 C 𝐶(100,200) 𝑦 ≤ 200 # of Hats 𝐵(20,200) B A 𝐴(20,100) D 𝐷(40,80) 𝑥 + 𝑦 ≤ 300 𝑦≥0 𝑥 + 𝑦 > 120 # of Foam Fingers FEASIBLE SOLUTIONS A point that is a possible answer to the overall question. Must be INSIDE the shaded region or ON A SOLID LINE If the point is on a dotted line at all, it is no longer feasible. Seek to find an optimal solution (either a maximum or a minimum of some quantity)…