chaoseverlasting said:I want to float another option. In this equation, [tex]abc+2fgh-af^2-bg^2-ch^2[/tex] is not 0. Therefore this curve represents a pair of straight lines. Geometrically this means that the pair of lines must always remain above the line y=1. The only way this is possible is if the pair of lines is parallel. If you find the equations of the two lines, then you should find that they lie above y=1.
An inequality proof is a mathematical technique used to demonstrate that one quantity is either greater or less than another quantity. This is typically done by manipulating equations or using logical arguments.
Inequality proofs are important in science because they allow us to make comparisons and draw conclusions about different quantities. This is especially useful in fields like economics, physics, and biology where understanding the relationships between different variables is crucial.
An inequality proof involves breaking down the problem into smaller, more manageable parts and then using logical arguments or mathematical equations to show how one quantity is greater or less than another. It is important to clearly state your assumptions and show all of your steps in the proof.
Sure! Here is a simple example: Prove that for any positive real numbers a and b, a + b > 2√(ab). First, we can square both sides to get (a + b)^2 > 4ab. Then, we can expand the left side to get a^2 + 2ab + b^2 > 4ab. Next, we can subtract 4ab from both sides to get a^2 - 2ab + b^2 > 0. This can be factored into (a - b)^2 > 0, which is always true for positive real numbers. Therefore, the original inequality is proven.
One common mistake is assuming that the inequality is true for all values of the variables, when in fact it may only be true for a certain range of values. It is important to clearly state any restrictions or assumptions in the proof. Additionally, it is important to check for any mathematical errors or logical fallacies in the proof before declaring it complete.