This Crash Course is all about optimization problems. Its interactive videos, key takeaways and review exercises will give you a quick refresher on the subject.
Before watching the videos of this Crash Course, be sure to know how to translate a statement into an inequality. You should also be able to graph inequalities on a Cartesian plane, and be familiar with the concepts of boundary lines and the polygon of constraints. You also need to master solving an inequality algebraically and solving a system of equations. These different notions will be essential for optimization problems.
Here are the steps to solve an optimization problem:
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Identify the variables.
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Translate the constraints into a system of inequalities.
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Establish the rule of the function to be optimized |(z).|
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Graph the polygon of constraints.
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Determine the vertices of the polygon of constraints.
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Find the optimal vertex (using a table or a scanning line).
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Give a complete answer.
To translate a constraint into an inequality, you need to find and interpret the keywords that represent inequalities.
| Symbol | Keywords |
|---|---|
|
|x<y| |
|x| is less than |y,| |x| is smaller than |y,| |x| is strictly less than |y,| etc. |
|
|x\leq y| |
|x| is less than or equal to |y,| |x| is at most equal to |y,| |x| does not exceed |y,| |x| is not more than |y,| etc. |
|
|x>y| |
|x| is greater than |y,| |x| is more than |y,| |x| exceeds |y,| |x| is strictly greater than |y,| etc. |
|
|x\geq y| |
|x| is greater than or equal to |y,| |x| is at least equal to |y,| |x| is a minimum of |y,| |x| is at least as much as |y,| etc. |
Tips and Tricks
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Each constraint produces one boundary line.
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It is best to always write the non-negative constraints, even if sometimes they are unnecessary.
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If the inequality sign of a constraint is |<| or |>,| then the boundary line is a dotted line.
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You can represent the solution set of inequalities using arrows or shading.
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If you can't determine the exact coordinates of the vertices of the polygon from the graph, you can calculate them using the comparison, substitution or elimination method.
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The optimal solution always corresponds to one of the vertices of the polygon of constraints, unless the vertex is situated on a dotted line, or the variables must be integers, but the vertex does not have integer coordinates.
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When establishing the rule for the function to be optimized, you need to specify if |z| is to be minimized or maximized.
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When you give the final answer, you need to write a complete sentence that takes the context into account. Your answer should include the values of |x| and |y| that optimize the function, and the maximum or minimum value obtained for the variable |z.|