Solving the Inequality ( frac{x^2 - 5x 6}{x^2 x 1} leq 0): A Step-by-Step Guide

When dealing with inequalities in mathematics, it is crucial to follow a systematic approach to find the solution set accurately. In this article, we will solve the inequality ( frac{x^2 - 5x 6}{x^2 x 1} leq 0 ). This guide is designed to help you understand the step-by-step process and provide valuable insights into solving such inequalities using various mathematical techniques.

Step 1: Factor the Numerator

The numerator of the given inequality is ( x^2 - 5x 6 ). Factoring this quadratic expression:

( x^2 - 5x 6 (x - 2)(x - 3) )

Step 2: Analyze the Denominator Using the Discriminant

The denominator is ( x^2 x 1 ). To analyze this, we can look at its discriminant:

( b^2 - 4ac 1^2 - 4 cdot 1 cdot 1 1 - 4 -3 )

Since the discriminant is negative, the quadratic ( x^2 x 1 ) has no real roots and is always positive (it opens upwards).

Step 3: Set Up the Inequality

Combining the results from the numerator and denominator, the inequality simplifies to:

( frac{(x - 2)(x - 3)}{x^2 x 1} leq 0 )

Since the denominator is always positive, the inequality is equivalent to finding where the numerator ( (x - 2)(x - 3) ) is less than or equal to zero.

Step 4: Find the Critical Points

The critical points are the roots of the numerator, which are ( x 2 ) and ( x 3 ). These points divide the number line into three intervals:

( (-infty, 2) ) ( (2, 3) ) ( (3, infty) )

Step 5: Test the Intervals

To determine where the product ( (x - 2)(x - 3) ) is negative, we test a point from each interval:

Interval ( (-infty, 2) ): Choose ( x 0 ) ( (0 - 2)(0 - 3) ( -2)(-3) 6 > 0 ) Interval ( (2, 3) ): Choose ( x 2.5 ) ( (2.5 - 2)(2.5 - 3) (0.5)(-0.5) -0.25 Interval ( (3, infty) ): Choose ( x 4 ) ( (4 - 2)(4 - 3) (2)(1) 2 > 0 )

Step 6: Conclusion

The product ( (x - 2)(x - 3) ) is negative in the interval ( (2, 3) ). Therefore, the solution to the inequality ( frac{x^2 - 5x 6}{x^2 x 1} leq 0 ) is ( 2 leq x leq 3 ).

[boxed{2 leq x leq 3}]

Additional Notes

It is important to note that when the denominator of a rational inequality is always positive or always negative, the signs of the inequality only depend on the numerator. In this case, the numerator being negative in the interval ( (2, 3) ) is the key to solving the problem.

For further practice and understanding of similar inequalities, feel free to explore more problems of this kind. Understanding the steps and techniques involved will help solidify your grasp on solving such mathematical challenges.