H. Algebra 2
9.3 Notes
9.3 Solving Rational Equations
Date: ____________
EXPLORE: Solving Rational Equations Graphically
A rational equation is an equation that contains one or more rational expressions. The time t in hours it
𝑑
takes to travel d miles can be determined by using the equation 𝑡 = 𝑟 , where r is the average rate of
speed. This equation is an example of a rational equation.
𝒙
One method to solve rational equations is by graphing. Solve the rational equation 𝒙−𝟑 = 𝟐 by
graphing.
A. Find the excluded values of x.
𝑥
B. Use the following table to find some specific points and graph the equation 𝑓(𝑥) = 𝑥−3.
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C. Graph the right side of the equation as 𝑔(𝑥) = 2
D. Find solution by finding the point of intersection.
E. Substitute solution back into original equation and verify.
This was just one of the ways we could have solved this equation graphically!
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H. Algebra 2
9.3 Notes
Solving Rational Equations Algebraically
Learning Target G: I can solve rational equations algebraically.
When solving rational equations, it is important to identify the excluded values in the original
equation, as some algebraic steps may change the equation resulting in these values being a solution.
When this happens, we have an ______________________________ solution. Extraneous solutions are not
solutions to the equation.
Solve each rational equation algebraically. Be sure to check for extraneous solutions.
A)
C)
3𝑥+7
𝑥−5
8
𝑥+3
=
=
5𝑥+17
2𝑥−10
B)
2𝑥−9
𝑥−7
𝑥
5
2
𝑥−7
+ =
𝑥+1
𝑥+6
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H. Algebra 2
9.3 Notes
Solving a Real-World Problem with a Rational Equation
Learning Target H: I can write a rational equation to model and solve a real-life problem.
Write a rational equation to solve the problem.
A) Kelsey is kayaking on a river. She travels 5 miles upstream and 5 miles downstream in a total of 6
hours. In still water, Kelsey can travel at an average speed of 3 miles per hour. What is the average
speed of the river’s current?
Analyze Information – Identify the important information




The answer will be the average speed of __________________________________________.
Kelsey spends ______ hours kayaking.
She travels ________ miles upstream and _______ miles downstream.
Her average speed in still water is ___________________________________.
Formulate a Plan




Let c represent the speed of the current in miles per hour.
When Kelsey is going upstream, her speed is equal to her speed in still water _____________ c.
When Kelsey is going downstream, her speed is equal to her speed in still water __________ c.
The variable c is restricted to _________________________________________________.

Complete the table
Distance (mi)
Upstream
5
Downstream
5
Average Speed (mi/h)
Time (h)
Use the results from the table to write an equation: Total time = time upstream + time downstream
Solve
Justify and Evaluate
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H. Algebra 2
9.3 Notes
B) Kevin can clean a large aquarium tank in about 7 hours. When Kevin and Lara work together, they
can clean the tank in 4 hours. Write and solve a rational equation to determine how much time it
would take Lara to clean the tank if she works by herself. Explain whether the answer is reasonable.
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