Inequality Proof: Is Multiplying Both Sides Valid?

In summary, the conversation is about the author's attempt to prove that multiplying a positive number by another positive number does not change the inequality. One person questions the validity of the proof, while the other argues that it is a valid assumption. The conversation also mentions the possibility of multiplication resulting in a negative number, which is not possible with two positive numbers.
  • #1
Bashyboy
1,421
5
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?
 
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  • #2
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.
 
  • #3
Shyan said:
I don't think for knowing this...

I am sorry, but I do not quite follow what you are saying here.
 
  • #4
Bashyboy said:
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.
 
  • #5


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.
 

1. Can you explain the concept of "multiplying both sides" in an inequality proof?

Multiplying both sides in an inequality proof means multiplying both the left and right sides of the inequality by the same number or variable. This is a valid operation as long as the number or variable is positive.

2. Why is it important to show the steps of multiplying both sides in an inequality proof?

Showing the steps of multiplying both sides in an inequality proof is important because it helps to demonstrate how the inequality remains true even after the operation is applied. This makes the proof more transparent and helps to avoid any errors or misunderstandings.

3. Is it always necessary to multiply both sides in an inequality proof?

No, it is not always necessary to multiply both sides in an inequality proof. In some cases, other operations such as addition, subtraction, or division may be used to prove the inequality.

4. Can you provide an example of when multiplying both sides is not valid in an inequality proof?

Multiplying both sides is not valid in an inequality proof when the number or variable being multiplied is negative. In this case, the direction of the inequality would need to be reversed in order for the proof to be valid.

5. How can I determine if multiplying both sides is valid in an inequality proof?

To determine if multiplying both sides is valid in an inequality proof, you need to ensure that the number or variable being multiplied is positive. You also need to make sure that the direction of the inequality remains the same after the multiplication is applied.

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