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30S Applied Math Mr. Knight – Killarney School Slide 1 Unit: Linear Programming Lesson: Intro to Linear Programming Intro to Linear Programming Learning Outcome B-1 LP-L1 Objectives: Review Linear Inequality concepts from earlier courses

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30S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Linear Programming Lesson: Intro to Linear Programming When you place a point on a line, the point creates three regions: the region that is the point itself the region to the left of the point the region to the right of the point Consider the number line for x = 5: The numbers to the right of 5 are larger than 5 and can be represented by the inequality x > 5, which is drawn on the number line below. Theory – Regions on a Line

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30S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Linear Programming Lesson: Intro to Linear Programming Many real-world situations involve ranges of values, as opposed to specific values. "Don't stay more than two hours in the shopping centre." "Practice your piano lessons for at least 45 minutes." "The hardware store will open from 7 a.m. till 10 p.m." "Jim has to spend four hours working in the yard every Saturday." In the last example, Jim has to mow the grass and clean up the yard. He probably spends some time daydreaming. It may be incorrect to say that mowing time + cleaning time = 4 h. It would be more appropriate to say mowing time + cleaning time + wasted time = 4 h. The process of dealing with variables that are ranges instead of specific values is called linear programming. It was developed during World War II by the United States Air Force to assist with planning and scheduling. Today, it is used by businesses to determine the maximum or minimum value of a specific expression, e.g., profit, time spent on a certain project, et cetera. Theory – Regions on a Line

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30S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Linear Programming Lesson: Intro to Linear Programming If the point, 5, is included in the set, the statement would be written x >= 5. In the diagram below, the point is shown as a solid dot, which means that it is included in the set. The numbers to the left of 5 are smaller than 5 and can be represented by the inequality x < 5, which is drawn on the number line below. If the point is included in the set, the statement would be written x 5. In the diagram below, the point is shown as a solid dot, which means that it is included in the set. Theory – Regions on a Line

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30S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Linear Programming Lesson: Intro to Linear Programming Graph the set of points represented by the statement: Test Yourself

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30S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Linear Programming Lesson: Intro to Linear Programming Graph the set of points represented by the statement: Test Yourself

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30S Applied Math Mr. Knight – Killarney School Slide 7 Unit: Linear Programming Lesson: Intro to Linear Programming Graph the set of points represented by the statement: Test Yourself

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30S Applied Math Mr. Knight – Killarney School Slide 8 Unit: Linear Programming Lesson: Intro to Linear Programming Graph the set of points represented by the statement: Test Yourself

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30S Applied Math Mr. Knight – Killarney School Slide 9 Unit: Linear Programming Lesson: Intro to Linear Programming Consider the inequality -x > -5. If we add 5 to each side and add x to each side, we get 5 > x. This means that 5 is larger than x. Therefore, x is less than 5. The set contains all numbers less than 5. If we multiply the original inequality, -x > -5, by -1, we get x > 5. This means that x is larger than 5. This is the opposite of the original and is incorrect. The direction of the inequality is wrong and has to be reversed. The correct operation is: Divide -x > -5 by -1 and reverse the direction of the inequality to get x -17. Theory – When to Reverse the Inequality

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30S Applied Math Mr. Knight – Killarney School Slide 10 Unit: Linear Programming Lesson: Intro to Linear Programming Graph the set of points represented by the statement: Test Yourself

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Graphing Linear Inequalities A linear inequality in two variables is an inequality that can be written in one of the following forms: A x + B y < C A x.

Graphing Linear Inequalities A linear inequality in two variables is an inequality that can be written in one of the following forms: A x + B y < C A x.

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