Lesson 22

This warm-up begins where students left off at the end of the previous lesson, considering how to multiply strategically to “clear the denominators” in a rational equation. The purpose of this warm-up is to elicit the idea that when multiplying to clear denominators, we only need to multiply by the least common denominator shared by the different terms in the expression. This idea will be used in the following activities as students solve rational equations and investigate extra solutions that sometimes arise in the solving process.

While students may notice and wonder many things about these images, the multiplication by \(x(x+2)\) and not \(x(x+2)^2\) (the result of multiplying together all the denominators) is the important discussion point.

Display the partially solved equation for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If students do not recall or have not heard the phrase “clearing the denominators” for this type of multiplying, tell them it is a useful description for the first few steps shown.

If time allows, have students finish solving for \(x\) by either using the quadratic formula or factoring (-1.5, 1).

The purpose of this activity is for students to understand how steps used to solve a rational equation sometimes lead to nonequivalent equations, giving rise to so-called extraneous solutions. For example, multiplying each side of a rational equation by \(x+1\) creates a new equation that is true when \(x= \text-1\), since \(0=0\), even if the original equation was not true at \(x= \text-1\).

Monitor for students who notice that neither 0 nor -1 can be substituted for \(x\) in the original expressions due to division by 0, and students who notice that multiplying by a variable on each side of an equation is different than the type of steps they have used in the past to solve linear or quadratic equations (specifically, that multiplying by \(x+1\) is the same as multiplying by 0 when \(x= \text-1\)).

The goal of this discussion is for students to share what they think is happening to make \(x= \text-1\) appear as a solution to the original equation even though it is not.

Invite previously identified students to share things they noticed about the original equation and what it means when we multiply each side of an equation by a variable. Consider these questions to help further the discussion:

• “Jada multiplied by \(x(x+1)\). What values of \(x\) make that expression equal to 0?” (0 and -1.)
• “What are values of \(x\) that we cannot substitute into the original equation?” (We cannot substitute 0 or -1 for \(x\) in the original equation, because both values create an invalid equation due to division by 0.)
• “What happens when you multiply each side of any equation by 0? What values of \(x\) make this new equation true but not necessarily the original equation?” (You get the equation \(0=0\), which is true for all values of \(x\).)

Help summarize the discussion by telling students that sometimes steps we do to solve an equation result in a new equation that is not equivalent to the original equation. Two equations are equivalent if they have the exact same solutions. Since \(x= \text-1\) is a solution to the new equation but not the original equation, some people call this an extraneous solution. The step that created a new equation that is not equivalent to the original equation was when each side was multiplied by an expression that can have the value 0. This can make two previously unequal (or undefined) sides equal for some value of the variable, so the original equation and the new equation are not equivalent. These inequivalent equations do not always arise from multiplying both sides of an equation by an expression, so we should always check the solution in the original equation to be sure. In later units, students will see other types of equation solving steps that can result in new equations that have solutions that do not satisfy the original equation.

The purpose of this activity is for students to practice solving rational equations and identifying extraneous solutions, if they exist. Students are expected to use algebraic methods to solve the equations and should be discouraged from using graphing technology.

Arrange students in groups of 2. Tell students to complete the first problem individually and then check their work with their partner before completing the following problems together.

Some students may forget how to solve a quadratic equation. Remind them of options such as factoring or using the quadratic formula when they have an expression equal to zero.

The purpose of this discussion is for students to share strategies for solving different types of equations. Some students may have thought that an equation was in the “least difficult” category, while others thought that the same equation was in the “most difficult” category. Remind students that once you feel confident about the strategies for solving an equation, it may move into the “least difficult” category, and recognizing good strategies takes practice and time.

Informally poll the class for each equation as to whether they placed it in the “most difficult” category, “least difficult” category, or if it was somewhere in the middle. Record and display the results for all to see.

For questions with a split vote, have a group share something that made it seem difficult about it and another group share something that made it seem less difficult. If there are any questions that everyone thought would be more difficult or everyone thought would be less difficult, ask students why it seemed that way. Ask students, “Were there any equations that were more difficult to solve than you expected? Were there any that were less difficult to solve than you expected?” If not brought up, ask how students worked out the equation \(\frac{2x+1}{x} = \frac{1}{x-2}\), which has irrational solutions.