Inequality Proof: Is Multiplying Both Sides Valid?

[Bashyboy]

Hello,

In Principles of Mathematical Analysis, the author is attempting to demonstrate that, if $x > 0$ and $y < z$, then $xy < xz$, which essentially states that multiplying by a positive number does not disturb the inequality.

I am hoping someone will quickly denounce this with an adequate explanation, but I feel as though the author is using the result to prove it.

He begins by noting that, if $z > y$, $z - y > 0$. He multiplies both sides by $x > 0$, and gets $x(z-y) > 0$.

This seems to be a special case of the theorem which we are trying to prove. Wouldn't this be an invalid step as we do not know what results from multiplying both sides of an inequality? Let $c = z - y$, and replace $y$ with zero in the theorem. This would give us

If $x > 0$ and $0 < c$, then $x \cdot 0 < xc$.

Am I mistaken?

 

[ShayanJ]

He is assuming that if you multiply two positive numbers, you will get a positive number. I don't think for knowing this, you need to know the theorem mentioned, so I see no problem with the proof.

 

[Bashyboy]

I don't think for knowing this...

I am sorry, but I do not quite follow what you are saying here.

 

[ShayanJ]

I am sorry, but I do not quite follow what you are saying here.

The product of two numbers is negative only when one of them is negative. You can't get a negative number by multiplying two positive numbers.

 

[CellCoree]

Hello,

Thank you for bringing up this interesting topic. I would like to provide a response to the question of whether multiplying both sides of an inequality is valid.

Firstly, it is important to note that the statement being discussed is a theorem, which means that it has been proven to be true using mathematical logic and reasoning. Therefore, we can trust that the author has followed a rigorous proof and has not made any invalid steps.

In mathematics, we often use known theorems and properties to prove new theorems. In this case, the author is using the distributive property of multiplication over addition to prove the theorem. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. This is a well-known and accepted property, so using it in a proof is valid.

Now, going back to your question about whether multiplying both sides of an inequality is valid. The answer is yes, as long as the number being multiplied is positive. This is because multiplying both sides of an inequality by a positive number will preserve the direction of the inequality. In other words, if a < b, then ac < bc, where c is a positive number. This is a fundamental property of inequalities and is widely accepted in mathematics.

In conclusion, the author's proof is valid and follows accepted mathematical principles. Multiplying both sides of an inequality by a positive number is a valid step in a proof, as long as the number is positive. I hope this explanation has helped clarify any doubts you may have had.