Inequality Proof: Max {A+B,C} ≤ Max {A,C} + Max {B,C}

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In summary, the inequality proof states that the maximum value of the sum of two numbers (A and B) and a third number (C) is always less than or equal to the sum of the maximum values of A and C, and B and C. This inequality is important in mathematics as it helps determine the maximum value among a set of numbers and is applicable in real-life situations such as economics and physics. An example of how this inequality can be proven is by using numbers such as A=5, B=10, and C=7. There are no exceptions to this inequality for real numbers, but it may not hold true for complex or imaginary numbers.
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stanley.st
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Hello,

i've met during problem solving with inequality

[tex]\max\{A+B,C\}\le\max\{A,C\}+\max\{B,C\}[/tex]

where A,B and C are real numbers. I don't know whether it holds, but I need to prove that.

Thanks for reply...
 
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  • #2
It holds if A,B,C are nonnegative, but play around with some negative numbers and you'll find a counterexample. (Hint: Does the inequality always hold if max{A+B,C}=A+B, or if max{A+B,C}=C? Pick the case you can't prove easily and search for counterexamples there.)
 

1. What is the meaning of "Inequality Proof: Max {A+B,C} ≤ Max {A,C} + Max {B,C}"?

The inequality proof states that the maximum value of the sum of two numbers (A and B) and a third number (C) is always less than or equal to the sum of the maximum values of A and C, and B and C.

2. Why is this inequality important in mathematics?

This inequality is important because it helps us determine the maximum value among a set of numbers. It is also a fundamental concept in understanding and solving more complex mathematical problems.

3. How can this inequality be applied in real-life situations?

This inequality can be applied in various fields such as economics, statistics, and physics. For example, it can be used to determine the maximum profit a company can make by calculating the maximum revenue from two different products and a fixed cost.

4. Can you provide an example of how this inequality can be proven?

Sure, let's say we have the numbers A=5, B=10, and C=7. The maximum value of A+B is 15, and the maximum value of A and C is also 15 (since A is already the maximum value). Similarly, the maximum value of B and C is also 15 (since B is already the maximum value). Therefore, the inequality holds true as 15 ≤ 15+15.

5. Are there any exceptions to this inequality?

No, this inequality holds true for all real numbers. However, it is important to note that the inequality may not hold true if the numbers are complex or imaginary.

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