Two equations are equivalent if they have the same solution set. The purpose of this lesson is for students to practice solving rational equations strategically and understand how certain steps in the solving process can lead to equations that are not equivalent to the initial equation. If a step in the solving process results in a new equation that has more solutions than the initial equation, people sometimes call these additional solutions extraneous solutions. One step that often results in an equation that is not equivalent is when each side of an equation is multiplied by an expression that can take on the value of 0, which could in turn make two previously unequal (or undefined) sides equal. In particular, this lesson gives students the chance to use precise language to explain how a solution is not always a solution due to multiplication by 0 (MP6).

This Warm-up begins where students left off at the end of the previous lesson, considering how to multiply strategically by paying attention to the structure of the equations (MP7) to “clear the denominators” in a rational equation. The first few activities help develop students’ understanding for how these extra, or extraneous, solutions can arise when solving rational equations. The last activity of the lesson provides practice for solving rational equations. Students first identify which equations they consider most and least difficult, and then select three to solve with a partner and check for possible extraneous solutions.