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How do you find points on a graph that are unclear due to them not being integers?

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1 Answer

When graphing a pair or system of inequalities, the resulting solution is the region of space occupied by both/all of the inequalities graphed.  The solution region may or may not have both a maximum and a minimum. 

If your solution region extends to the top of the graph, there is no maximum. If your solution region extends to the bottom of the graph there is no minimum.  (Beware: you may have to imagine a larger graph than the one you have drawn )

If the region has a clear maximum or minimum shown on your graph, it will occur at the intersection of the linear borders of the graphed inequalities.

Your question, as I understand it, is how to determine the coordinates of this intersection point if it doesn't land on integers. There's really two ways: one approximate and the other exact.

Approximate solution: This can only give you a close answer and is dependent on your eyeball judgement.

1. Note the two x integers surrounding the intersection.

2. Estimate what percentage of the way from the lesser interger to the greater integer is the point.  You can estimate by about a half: *.5   About a third or two thirds *.3, *.6. About a quarter or three quarters *.2, *.7. 

3. Do the same thing in the y direction.

Exact solution: This will give you the exact answer even if it is a messy fraction.

1. Write the two equations that represent the linear borders of the inequalities you graphed. (This just means changing the inequalitiy sign to an equal sign.)

2. Solve the system of two equations for x and y. You can use linear combination or substitution.

And a word to the wise: if you want to check your accuracy in solving the system, do both methods, but report the exact answer when it is close to your eyeball estimate. If it seems very different than an eye estimate, you should recheck your algebra when you solved. There's probably a silly error someplace that threw it off. Using estimates to check your answers is a great way to catch your own mistakes.