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)

#Algebra

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

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 Jain - 5 years ago

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 - 5 years ago

True that.....

Rishabh Jain - 5 years ago

@Rishabh Jain – Please edit your "Note", especially since it is in bold. Thanks!

Calvin Lin Staff - 5 years ago

@Calvin Lin – But can we even subtract and divide two inequalities directly.. ??

Rishabh Jain - 5 years ago

@Rishabh Jain – 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 - 5 years ago

@Calvin Lin – Thanks.. Added

Rishabh Jain - 5 years ago

@Calvin Lin

Aditya Dhawan - 5 years ago

@Rishabh Cool

Aditya Dhawan - 5 years ago

@Nihar Mahajan

Aditya Dhawan - 5 years ago

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 - 5 years ago

Sorry for the typo. I have edited the note

Aditya Dhawan - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$