Solving the Inequality ( frac{x-1}{x^2} leq 1 ) with Steps and Solutions

In this article, we will explore the step-by-step solution for the inequality ( frac{x-1}{x^2} leq 1 ). We will break down the problem using the definition of absolute value and analyze the inequality in two different cases.

Rewriting the Inequality

First, let's rewrite the given inequality:

( frac{x-1}{x^2} leq 1 )

We can rewrite this as:

( frac{x-1}{x^2} - 1 leq 0 )

This simplifies to:

( frac{x-1 - x^2}{x^2} leq 0 )

Further simplification yields:

( frac{-3 - x^2}{x^2} leq 0 )

Case 1: ( x - 1 geq 0 ) (i.e., ( x geq 1 ))

In this case, ( x - 1 x - 1 ). The inequality becomes:

( frac{x - 1 - x^2}{x^2} leq 0 )

This can be further simplified to:

( frac{-3 - x^2}{x^2} leq 0 )

For this inequality to hold true, the denominator must be positive, which occurs when:

( x^2 > 0 )

Given that we are assuming ( x geq 1 ), the solution for this case is:

( x geq 1 )

Case 2: ( x - 1

In this case, ( x - 1 1 - x ). The inequality becomes:

( frac{1 - x}{x^2} - 1 leq 0 )

This simplifies to:

( frac{1 - x - x^2}{x^2} leq 0 )

Further simplification yields:

( frac{-2x - 1}{x^2} leq 0 )

To determine the intervals where this inequality holds true, we analyze the critical points:

Numerator: ( -2x - 1 0 ) implies ( x -frac{1}{2} )

Denominator: ( x^2 0 ) implies ( x 0 )

Now, let's test the intervals defined by these critical points:

For ( x

( frac{-2(-3) - 1}{(-3)^2} frac{6 - 1}{9} frac{5}{9} > 0 )

For ( -2

( frac{-2(-1) - 1}{(-1)^2} frac{2 - 1}{1} 1 > 0 )

For ( -frac{1}{2}

( frac{-2(0) - 1}{(0)^2} ) is undefined (but the test interval includes ( 0 ), so we consider the limit behavior which makes the expression negative)

For ( 0

( frac{-2(frac{1}{2}) - 1}{(frac{1}{2})^2} frac{-1 - 1}{frac{1}{4}} -8

Therefore, the solution for this case is:

( -infty

Final Solution

Combining the results from both cases, the solution to the inequality ( frac{x-1}{x^2} leq 1 ) is:

( (-infty, -2) cup left(-frac{1}{2}, 1right] cup [1, infty) )

Conclusion

This article has provided a detailed breakdown of solving the inequality ( frac{x-1}{x^2} leq 1 ). The key steps include breaking down the inequality, applying the definition of absolute value in two different cases, and determining the intervals where the inequality holds true. Understanding these methods can help in solving similar inequalities in the future.