Unit 3, Lesson 7
Solving Rational Equations
Algebra 2
7.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-REI.A.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Building Toward
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
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 (MP7). 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 and not (the result of multiplying together all the denominators) is the important discussion point.
Launch
Arrange students in groups of 2. Display the equations 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 with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the equations. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If time allows, ask students to finish solving for by either using the quadratic formula or factoring. ()
7.2
Activity
Instructional Routines
MLR1: Stronger and Clearer Each TimeMaterials
NoneActivity Narrative
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 creates a new equation that is true when , since , even if the original equation was not true at .
Monitor for students who notice that neither 0 nor -1 can be substituted for 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 from the type of steps students have used in the past to solve linear or quadratic equations (specifically, that multiplying by is the same as multiplying by 0 when ).
7.3
Activity
Materials
NoneActivity Narrative
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.
Launch
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.
MLR8 Discussion Supports. Display sentence frames to help students describe their reasons for choosing values of that cannot be solutions to the equations. Display sentence frames to support whole-class discussion. Examples:
- “ and cannot be -values because .”
- “I noticed , so I know .”
Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to identify 2 equations that they think would be least difficult to solve and 2 that would be most difficult to solve and then choose and respond to at least 2 of the questions they identified.
Supports accessibility for: Organization, Attention
Supports accessibility for: Organization, Attention
Lesson Synthesis
The purpose of this writing prompt is for students to reflect on what they have learned about how so-called extraneous solutions can arise when solving rational equations. Ask students to respond to the following prompt: “How can extra solutions arise in the process of solving an equation?” Encourage students to use equations from the lesson to help in their explanations if needed.
If time allows, ask students to share what they’ve written with a partner and then select 2–3 students to share with the class something from either their or their partner’s paper. These are two key understandings: When we multiply each side of an equation by an expression that can have the value 0, we sometimes get an equation that is not equivalent to the initial equation. And we should always check to see if a solution satisfies the original equation.
Student Lesson Summary
Consider the equation . We could solve this equation for by multiplying each expression by to get an equation with no variables in denominators, and then rearranging it into an expression that equals 0. Here is what that looks like:
The last equation, , leads us to believe that the original equation has two solutions: and . Substituting into the original equation, we get , which is true since each side is equal to . But, substituting into the original equation, we get , which isn’t a valid equation since division by 0 is not allowed. This means isn’t a solution, so what happened to make us think that it was?
Let’s consider the simpler equation . This equation has one solution, . But if we multiply each side by the result is a new equation, , which has solutions 5 and 1. The 1 is a solution to the new equation because when , . But if we substitute 1 for into the original equation, we get , which is not a valid equation, so 1 is not a solution to the original equation. Because we multiplied each side of the original equation by an expression that has the value 0 when , the two sides and 0 that were unequal at that specific -value are now equal. For this example, is sometimes called an extraneous solution.
In the original example, is the extraneous solution. While is a solution to the equation we wrote after we multiplied the original equation by on each side, it is not a solution to the original equation since they are not equivalent. It should be noted that even though we multiplied by , , and , only one extraneous solution was added. This shows that multiplying by an expression that can equal 0 does not always cause an extraneous solution. So how do we tell if a solution is extraneous or not? We substitute it into the original equation and make sure the result is a valid equation.
- When multiplying each side of an equation by an expression with a variable, we identify what values would make the expression 0 and rule those out as possible solutions.
- Once solutions are found, we substitute them into the original equation and make sure the result is a valid equation.