Mastering Rational Inequalities: Steps and Solutions

Miacademy & MiaPrep Learning Channel

This instructional video provides a comprehensive guide to solving rational inequalities, a core topic in advanced algebra. The lesson begins by distinguishing between rational equations, which typically have a finite number of solutions, and rational inequalities, which possess infinite solutions within specific intervals. The narrator introduces a structured four-step process for solving these problems: altering the inequality to set one side to zero, factoring to identify boundary points, testing intervals on a number line, and writing the final solution set using inequality notation. The video walks through two detailed examples that cover different scenarios. The first example demonstrates how to handle an inequality that is already set to zero, focusing on factoring quadratics and determining whether boundary points are included (closed circles) or excluded (open circles). The second, more complex example shows how to manipulate an inequality with fractions on both sides by finding a common denominator and combining terms before solving. The narrator emphasizes the critical rule that values making the denominator zero are never included in the solution set. Teachers can use this video to introduce the algebraic method for solving rational inequalities or as a review tool for students struggling with sign analysis. The visual representation of the number line and the step-by-step testing of points helps demystify the abstract concept of solution intervals. The clear distinction between numerator roots (which can be included) and denominator roots (which are always excluded) addresses a common student misconception directly.