What does adding two inequalities represent?

Question

Let's say I have two inequalities. Now when I add them, I get a third inequality. Is every solution to this inequality a solution to both the original inequalities, either one of the two or neither? What does adding two inequalities represent?

Answer 1

A solution of a sum of inequalities is a solution of one of the original inequalities. Otherwise we'd have

$$ a >b $$ $$ c > d$$ Therefore $ a+c>b+d$, a contradiction. This is the only conclusion you can draw.

Answer 2

I think we can understand it by the following way.

$a>b$ says $a-b>0$.

$c>d$ says $c-d>0$.

Sum of positive numbers is a positive number.

Thus, $a-b+c-d>0$ or $a+c-(b+d)>0$, which says $a+c>b+d$ and we are done!

The second part of your statement is wrong.

For example.

Let $x>3$ and $x>5$.

After summing we get $x>4$ and for $4.5$ we see that $x>5$ is wrong.

Answer 3

i am considering $$2x+3>1\; \text{and}\; x+1>2$$ these two inequalities represent area not including boundary and now their sum $$4x+4>3$$ http://www4f.wolframalpha.com/Calculate/MSP/MSP427821017e477697d9e60000120c365d0dgge811?MSPStoreType=i I can,t copy the other links so you can understand by reading the graphs that what each inequality represent............ but each inequality represents different and their sum also....