## Introduction to Mathematical Inequalities

A deep understanding of mathematical inequalities is crucial in numerous fields, including algebra, calculus, and real analysis. Today, we'll explore the nuances of a specific inequality and how to simplify and solve it. We'll also highlight the importance of checking the domain and ensuring our solutions are valid.

Understanding the Initial Equation

Let's start by examining the expression:

[ left| frac{-6n-5}{n_1n_2} right| frac{6n^5}{n_1n_2} ]

This equation is only true for specific values of n. We need to consider the signs of the terms in the expression. For the absolute value to be equivalent to the given fraction, the fraction must be negative so that it is multiplied by -1.

Determining the Conditions for (n)

There are two conditions to consider:

(-6n - 5) and (n_1n_2) are both negative.

(-6n - 5) and (n_1n_2) are both positive.

If the fraction is to be negative, we need:

(-6n - 5

This simplifies to:

[ n > -frac{5}{6} ] and [ n

or

[ n > -frac{5}{6} ] and [ n > -1 ]

Combining the conditions, we get:

[ -2

If (n in mathbb{Z}^ ) (positive integers), then only ( n 1) is a valid solution.

Checking the Second Inequality

Now, let's focus on the second part of the problem and simplify the inequality:

[ frac{6n^5}{n_1n_2} leq frac{6n^6}{n_1n_2} ]

We can simplify this by multiplying both sides by (n_1n_2) (assuming (n_1n_2 > 0)):

[ 6n^5 leq 6n^6 ]

Divide both sides by 6:

[ n^5 leq n^6 ]

For positive (n), we can further simplify this to:

[ 1 leq n ]

Therefore, ( n geq 1 ).

Conclusion and Final Check

We now have the combined solution:

[ n 1 ] or [ -2

Remember that for the second part of the inequality, (n) must be a positive integer, ensuring that (n 1) is the final valid solution.

## Summary of Key Points

Check the domain of the variables involved.

Simplify the inequalities by isolating the variables.

Ensure all steps are valid and the final solution is consistent with the initial conditions.

By following these steps, you can solve complex mathematical inequalities systematically and accurately.