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Muti-variable system of linear inequalties

We are all aware of the usual graphical method of solving a multi-variable system of inequalities. But how could one solve such a system algebraically? Also, does a system of inequalities follow the usual arithmetic of simultaneous equations? For instance, does the system \[ x+y>3 \\ x+2y>5 \]

necessarily imply \(y>2\). ( It does not seem to, since \(x=100\) and \(y=1\) is a perfectly valid solution)

Note by Aditya Dhawan
3 months, 3 weeks ago

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No, algebra of inequalities is not that simple... You cannot just simply subtract or divide two inequalities like we do to two equations. As its evident that you have subtracted inequalitities which isn't allowed and hence gave wrong results..... As an explicit example, if it had been:

\[x+y>3~~\text{and}~~ x+2y<5\] Then multiply first equation by \((-1)\) and add to second equation to get:

\[-(\not x+y)+\not x+2y<2\iff y<2\]

Note:- You can only add two inequalities​ together. You will not always be able to multiply, subtract or divide them.

PS:- I don't know any algebraic method to solve system these type of linear equations in two variables as I always relied on graph for getting solutions and they did the job fairly easily! Rishabh Cool · 3 months, 2 weeks ago

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@Rishabh Cool Note: You cannot multiply two inequalities. For example, \( -2 < 1 \) and \( -2 < 1 \). Can we multiply them to conclude that \( 4 < 1 \)?

What is true is that you can multiply and divide by terms with the same sign. IE You can multiply/divide by \( - (x+y)^2 \), but not by \( -x+y \). Calvin Lin Staff · 3 months, 2 weeks ago

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@Calvin Lin True that..... Rishabh Cool · 3 months, 2 weeks ago

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@Rishabh Cool Please edit your "Note", especially since it is in bold. Thanks! Calvin Lin Staff · 3 months, 2 weeks ago

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@Calvin Lin But can we even subtract and divide two inequalities directly.. ?? Rishabh Cool · 3 months, 2 weeks ago

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@Rishabh Cool No, you cannot even multiply two inequalities. As stated,

What is true is that you can multiply and divide by terms with the same sign.

So, your note should be

Note:- You can only add two inequalities​ together. You will not always be able to multiply, subtract or divide them

Calvin Lin Staff · 3 months, 2 weeks ago

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@Calvin Lin Thanks.. Added Rishabh Cool · 3 months, 2 weeks ago

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@Nihar Mahajan Aditya Dhawan · 3 months, 2 weeks ago

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@Rishabh Cool Aditya Dhawan · 3 months, 2 weeks ago

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@Calvin Lin Aditya Dhawan · 3 months, 2 weeks ago

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If I've understood what you mean I think you could divide this system in 3 disequalities (a>b>c --> a>b, b>c, a>c) but I don't know if it works. Anyway with the values x=100 and y=1 you obtain 101 > 302 > 5, that is not true. Matteo Monzali · 3 months, 2 weeks ago

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@Matteo Monzali Sorry for the typo. I have edited the note Aditya Dhawan · 3 months, 2 weeks ago

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