Proving The Inequality 12x^2 + 10y^2 < 24xy Under Given Conditions

Introduction

In this article, we will delve into the proof of an inequality involving positive real numbers x and y. The problem states that given x - 4y < y - 3x, and assuming 3x > 2y, we aim to prove that 12x² + 10y² < 24x**y. This exploration requires a careful manipulation of the given inequalities and a strategic approach to arrive at the desired conclusion. The inequality problems are fundamental in mathematical analysis and often appear in various competitions and research contexts. Understanding the techniques involved in solving such problems is crucial for developing a deeper understanding of mathematical concepts and problem-solving skills. The journey to proving this inequality will not only demonstrate the elegance of mathematical logic but also highlight the importance of algebraic manipulation and inequality properties. We will explore how the initial conditions interplay to allow us to make specific deductions, ultimately leading to the establishment of the target inequality. By breaking down the problem into manageable steps and providing clear explanations, this article aims to be an accessible resource for anyone interested in mathematical problem-solving.

Problem Statement and Initial Conditions

To begin, let's restate the problem clearly. We are given that x and y are positive real numbers satisfying the inequality x - 4y < y - 3x. Additionally, we are given the condition 3x > 2y. Our objective is to prove that 12x² + 10y² < 24x**y. These initial conditions provide the foundation upon which we will build our proof. The first inequality, x - 4y < y - 3x, is a crucial starting point. Simplifying this inequality will reveal a fundamental relationship between x and y, which we will leverage throughout the proof. The second condition, 3x > 2y, further refines the possible values of x and y and will likely play a key role in the later stages of the proof. Understanding how these conditions interact is essential for devising a successful strategy. We need to carefully analyze each condition and determine how it contributes to the overall goal of proving the target inequality. This initial assessment sets the stage for the subsequent algebraic manipulations and logical deductions. By meticulously examining the problem statement and identifying the key relationships, we can approach the proof with a clear direction and a higher chance of success.

Simplifying the Given Inequality

The first step in our proof is to simplify the inequality x - 4y < y - 3x. By adding 3x and 4y to both sides of the inequality, we get:

x - 4y + 3x + 4y < y - 3x + 3x + 4y

This simplifies to:

4x < 5y

This simplified inequality, 4x < 5y, provides a crucial relationship between x and y. It tells us that x is less than 5/4 times y. This relationship will be instrumental in our subsequent steps, as it allows us to substitute or compare terms in a meaningful way. The simplification process is a fundamental technique in mathematical problem-solving. By reducing complex expressions to their simplest forms, we can often reveal hidden relationships and make the problem more tractable. In this case, simplifying the initial inequality has given us a clear and concise relationship between x and y that we can directly use in our proof. This step underscores the importance of algebraic manipulation in solving inequalities. By applying basic algebraic operations, we have transformed the original inequality into a more manageable form, paving the way for further progress. The inequality 4x < 5y is not just an intermediate result; it is a key piece of the puzzle that will help us connect the given conditions to the inequality we aim to prove.

Utilizing the Condition 3x > 2y

We are also given that 3x > 2y. This condition, combined with the simplified inequality 4x < 5y, provides us with a range of possible values for x and y. Specifically, we have two inequalities involving x and y: 4x < 5y and 3x > 2y. These inequalities will be pivotal in constructing our proof. The condition 3x > 2y gives us a lower bound on the ratio of x to y, while 4x < 5y gives us an upper bound. Together, they constrain the possible relationships between x and y, which is essential for our proof strategy. Inequalities, in general, provide bounds and constraints on variables, which is a powerful tool in mathematical analysis. By leveraging these constraints effectively, we can narrow down the possibilities and establish the desired result. In this case, the two inequalities, 4x < 5y and 3x > 2y, work together to limit the possible values of x and y, making it easier to prove the target inequality. The strategic use of these conditions is at the heart of our proof approach. We will explore how to combine these inequalities with algebraic manipulations to ultimately demonstrate that 12x² + 10y² < 24x**y.

Constructing the Proof

Now, let's focus on proving the inequality 12x² + 10y² < 24x**y. To do this, we can try to manipulate the left-hand side of the inequality and show that it is indeed less than the right-hand side. A common technique in proving inequalities is to work backward from the desired result or to try to rewrite the expression in a more convenient form. In this case, let's consider the difference between the right-hand side and the left-hand side:

24x**y - (12x² + 10y²) = -12x² + 24x**y - 10y²

Our goal is to show that this difference is positive. If we can demonstrate that -12x² + 24x**y - 10y² > 0, then we will have proven the original inequality. This approach of considering the difference is a standard technique in inequality proofs. By focusing on the difference, we transform the problem into showing that an expression is either positive or negative. This can often simplify the analysis and make it easier to apply known inequalities or algebraic manipulations. In this case, we will focus on showing that -12x² + 24x**y - 10y² is positive. This expression involves both x² and y², as well as the cross-term x**y. To make progress, we need to find a way to relate this expression to the inequalities we have already derived: 4x < 5y and 3x > 2y. The challenge lies in finding the right combination of algebraic manipulations and substitutions that will allow us to establish the desired inequality.

Completing the Square (Author's Hint)

As hinted by the author, let's try to complete the square. We have the expression -12x² + 24x**y - 10y². To complete the square, we can try to rewrite this expression in the form -a(x - by)² + cy² for some constants a, b, and c. Completing the square is a powerful technique for analyzing quadratic expressions. It allows us to rewrite a quadratic expression in a form that reveals its minimum or maximum value and its roots. In the context of inequalities, completing the square can help us show that an expression is always positive or always negative, which is often the key to proving the inequality. In this case, we have a quadratic expression in two variables, x and y. Completing the square will involve manipulating the expression to group the terms involving x into a perfect square, while the remaining terms will involve y². This will allow us to analyze the expression more easily and relate it to the inequalities we have already derived. The hint to complete the square is a crucial piece of guidance in this problem. It suggests a specific path that will lead to the solution. Without this hint, it might be difficult to see how to proceed from the previous step. The technique of completing the square requires careful algebraic manipulation, but it is a valuable skill to have in one's mathematical toolkit. Let's proceed with the completion of the square to see how it helps us prove the inequality.

Rewriting the Expression

Let's rewrite the expression -12x² + 24x**y - 10y² by factoring out -12 from the terms involving x:

-12(x² - 2x**y) - 10y²

Now, to complete the square inside the parentheses, we need to add and subtract (y)²:

-12(x² - 2x**y + y² - y²) - 10y²

This can be rewritten as:

-12((x - y)² - y²) - 10y²

Expanding, we get:

-12(x - y)² + 12y² - 10y²

Which simplifies to:

-12(x - y)² + 2y²

This transformation is a crucial step in completing the square. By adding and subtracting y² inside the parentheses, we have created a perfect square trinomial, x² - 2x**y + y², which can be factored as (x - y)². This allows us to rewrite the original expression in a form that is more amenable to analysis. The technique of adding and subtracting the same term is a common trick in algebraic manipulation. It allows us to rewrite expressions without changing their value, but in a way that makes them more useful for our purposes. In this case, adding and subtracting y² has allowed us to complete the square and isolate the term (x - y)². The resulting expression, -12(x - y)² + 2y², is a significant simplification of the original expression. It clearly shows the dependence on (x - y)² and y², which will be instrumental in the next steps of our proof. This rewriting process highlights the power of algebraic manipulation in simplifying complex expressions and revealing hidden structures.

Analyzing the Rewritten Expression

We now have the expression -12(x - y)² + 2y². To prove that this is positive, we need to show that 2y² > 12(x - y)². Dividing both sides by 2, we get:

y² > 6(x - y)²

Taking the square root of both sides (since both sides are positive), we get:

y > √6 |x - y|

This inequality is the key to proving our original inequality. We have transformed the problem into showing that y is greater than √6 times the absolute value of (x - y). This transformation is a significant step forward. We have reduced the problem to a more manageable inequality involving y and the absolute value of (x - y). The square root operation is justified because both sides of the inequality are positive. This ensures that the inequality sign is preserved. The introduction of the absolute value is important because (x - y) can be either positive or negative. The absolute value ensures that we are dealing with a positive quantity. The inequality y > √6 |x - y| provides a direct relationship between y and the difference between x and y. To prove this inequality, we will need to leverage the given conditions 4x < 5y and 3x > 2y. The challenge now is to show that these conditions imply that y is indeed greater than √6 times the absolute value of (x - y). This step highlights the importance of careful algebraic manipulation and the use of properties of inequalities.

Using Given Conditions to Finalize the Proof

To prove y > √6 |x - y|, we consider two cases:

Case 1: x ≥ y

In this case, |x - y| = x - y, so we need to show y > √6 (x - y). This is equivalent to (√6 + 1)y > √6 x, or y > (√6 / (√6 + 1))x. We know 4x < 5y, which implies y > (4/5)x. We need to show (4/5)x > (√6 / (√6 + 1))x, which is equivalent to 4/5 > √6 / (√6 + 1). This simplifies to 4(√6 + 1) > 5*√6*, or 4 > √6, which is true.

Case 2: x < y

In this case, |x - y| = y - x, so we need to show y > √6 (y - x). This is equivalent to (√6 - 1)y < √6 x, or y < (√6 / (√6 - 1))x. We know 3x > 2y, which implies y < (3/2)x. We need to show (3/2)x > (√6 / (√6 - 1))x, which is equivalent to 3/2 > √6 / (√6 - 1). This simplifies to 3(√6 - 1) > 2*√6*, or √6 > 3, which is false. However, we also know 4x < 5y which mean that x < (5/4)y. Combining this with 3x > 2y, we get (2/3)y < x < (5/4)y. Thus x-y < (5/4)*y-y = (1/4)y. We need y > √6(y-x), thus 1 > √6(1-x/y). since x/y < 1, 1-x/y > 0. Since x > (2/3)y, thus 1-x/y < 1/3. So we need 1 > (√6)/3, or 3 > √6, which is true.

In both cases, we have shown that y > √6 |x - y|, which means -12(x - y)² + 2y² > 0, and therefore 12x² + 10y² < 24x**y.

Conclusion

In this article, we successfully proved the inequality 12x² + 10y² < 24x**y under the given conditions x - 4y < y - 3x and 3x > 2y, where x and y are positive real numbers. The proof involved simplifying the initial inequality, utilizing the condition 3x > 2y, completing the square, and analyzing the rewritten expression. By considering different cases and leveraging the given conditions, we were able to establish the desired inequality. This problem demonstrates the importance of algebraic manipulation, inequality properties, and strategic problem-solving in mathematical proofs. The technique of completing the square played a crucial role in simplifying the expression and revealing the underlying structure. The careful analysis of the cases x ≥ y and x < y allowed us to handle the absolute value and complete the proof. The successful solution to this problem highlights the power of mathematical reasoning and the beauty of logical deduction. The steps taken in this proof serve as a valuable example of how to approach inequality problems and can be applied to a wide range of similar challenges.