How to Prove x ≤ xy/2 ≤ y for Real Numbers x and y

Introduction

This article aims to prove a fundamental inequality in real numbers: x ≤ xy/2 ≤ y given that x y. This involves a detailed step-by-step approach to understanding and proving the given inequality, along with its applications in trigonometric forms.

Proof of the Inequality

Given that x y, we need to prove that x ≤ xy/2 ≤ y.

Step 1: Proving x ≤ xy/2

First, let's consider the expression xy/2.

xy/2 x/2 y/2

Since it is given that x y, it follows that x/2 y/2. Therefore, adding the same terms to both sides, we get:

x/2 x/2 y/2 y/2

Which simplifies to:

x y/2 x/2

Substituting xy/2 x/2 y/2, we get:

x xy/2

To prove the equality part, we need to show that x xy/2 when x y. This is because both sides would then be equal to x y. Hence, the equality holds when x y but otherwise, x xy/2 is true.

Step 2: Proving xy/2 ≤ y

Now, to prove that xy/2 ≤ y, we can start from the expression:

xy/2 x/2 y/2

Since we know that y x/2 (from the previous inequality), then adding y/2 to both sides, we get:

y/2 y/2 ≥ x/2 y/2

This simplifies to:

y ≥ xy/2

Therefore, we have shown that:

x ≤ xy/2 ≤ y

This completes the proof for the inequality.

Mathematical Notation and Proof Using Inequalities

Formally, we can write the proof as follows:

Given: x y

Then:

x ≤ xy/2 ≤ y

Mathematically, this can be expressed as:

x y Rightarrow; x ≤ xy/2 ≤ y

Similarly, we can prove the other part:

xy/2 x/2 y/2

x/2 y/2 Rightarrow; x/2 x/2 y/2 y/2 Rightarrow; x xy/2

And

x/2 y/2 Rightarrow; x/2 y/2 ≥ y/2 y/2 Rightarrow; xy/2 ≥ y

Hence, we conclude:

x ≤ xy/2 ≤ y

Trigonometric Application Example

Given that x cos^(-1)α and y sin^(-1)α, the inequality x ≤ xy/2 ≤ y can be further explored.

Using these conditions, we can substitute:

xy cos^(-1)α sin^(-1)α

From the given condition, we get:

1/√2 ≤ α ≤ 1

Then, we calculate:

xy/2 (cos^(-1)α sin^(-1)α) / 2

Similarly:

xy/2 (π/2) / 2 π/4

Substitution back into the inequality, we obtain:

cos^(-1)α ≤ π/4 ≤ sin^(-1)α

This result is valid under the given conditions and provides an interesting application of the original inequality in a trigonometric context.

Conclusion

In this article, we have proven the inequality x ≤ xy/2 ≤ y for real numbers x, y, and explored its application in a trigonometric context. The proof involves detailed step-by-step analysis and the use of basic inequality properties. Understanding such inequalities is crucial in various fields, including mathematics and applications in real-life scenarios.