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Here are some examples: The easiest way to graph this is with the “cover up” or intercepts method, since we have variables on one side and the constant on the other.To graph using the intercepts method, cover up the \(4y\) (making \(y=0\)) and we see the \(x\) intercept is , since \(-4\times -3=12\).Finally, we substitute these ordered pairs into our objective equations and select the maximum or minimum value, based on the desired result.
Here are what some typical Systems of Linear Inequalities might look like in Linear Programming: We’ll graph the equations as equalities, and shade “up” or shade “down”.
The easiest way to graph the first two inequalities is with the intercepts or “cover up” method.
To remember this, I think about the fact that “\(\)” have less pencil marks than “\(\le\)” and “\(\ge\)”, so there is less pencil used when you draw the lines on the graph.
You can also remember this by thinking the line under the “\(\le\)” and “\(\ge\)” means you draw a solid line on the graph.
It usually involves a system of linear inequalities, called constraints, but in the end, we want to either maximize something (like profit) or minimize something (like cost).
Whatever we’re maximizing or minimizing is called the objective function.
Note that the last example is a “Compound Inequality” since it involves more than one inequality.
The solution set is the ordered pairs that satisfy both inequalities; it is indicated by the darker shading.
So this means we need to find a way of using these limitations to our advantage, like finding the optimum amount of money, space, time, etc., to accomplish our goals. First, we must identify all constraints, by creating a system of inequalities.
Then we must identify the Objective Function, which is the equation we want to maximize or minimize.