Why is it useful to multiply a rational inequality by the square of the denominator?

Multiplying a rational inequality by the square of the denominator is particularly beneficial because it ensures positivity in the inequality. When you use the square of the denominator, you are guaranteed that this term will always be positive, regardless of the values of the variable involved. This is crucial in inequalities because they can change direction when one side becomes negative. By multiplying through by a positive quantity, the relationship represented by the inequality remains intact.

Moreover, ensuring that the expression remains positive removes ambiguity that might arise from restrictions in the domain, such as points where the denominator is zero or negative. Since the square of any real number is non-negative, this approach helps maintain the integrity of the inequality, allowing for a straightforward analysis of the inequality without worrying about sign changes that might complicate the solution process.