Section 6.5 Rational Equations and Models
Objective 2: Solve Rational Equations
Definition Rational Equation
A rational equation results when we set two rational expressions equal to each other. Some
examples of rational equations are
x  2 1 3x  4
 
x7 x
x 1
and
x2  3x1  4 .
Solving Rational Equations
Step 1.
List all restricted values (values that make any denominators equal zero).
Step 2.
Determine the LCD of all denominators in the equation.
Step 3.
Multiply both sides of the equation by the LCD.
Step 4.
Solve the resulting polynomial equation.
Step 5.
Discard any restricted values and check the remaining solutions in the original
equation.
6.5.7 Solve the rational equation. If there is no solution, state this as your answer.
6.5.11 Solve the rational equation. If there is no solution, state this as your answer.
6.5.14 Solve the rational equation. If there is no solution, state this as your answer.
A special case of Rational Equations is a proportion.
A proportion is a statement that two ratios are equal, such as
P R
 .
Q S
When we have a rational equation that is a proportion, we can begin by cross-multiplying to
get P  S  Q  R and then solve the resulting polynomial equation.
Cross-Multiplying
P R

Q S
PS  Q R
Note that any solutions that are restricted values in the original equation must be discarded.
6.5.15 Solve the rational equation. If there is no solution, state this as your answer.
Objective 3: Find the zeros of a Rational Function
Let R  x  be a rational function. If c is a real number in the domain of R such that R  c   0 , then c
is called a zero, or root, of the rational function. The zeros of a rational function are the solutions to
the equation R  x   0 .
Finding the Zeros of a Rational Function
To find the zeros, or roots, of a rational function of the form
P  x
, do the following:
Q  x
Step 1. Find the restricted values by setting the denominator equal to zero and solving the resulting
equation, Q  x   0 .
Step 2. Set the numerator equal to zero and solve the resulting equation, P  x   0 .
Step 3. Any solution from step 2 that is not a restricted value is a zero of the function.
6.5.17 Find the zeros of the rational function.
Objective 4: Use Rational Equations to Solve Application Problems
Follow the usual problem-solving strategy when solving application problems involving rational
equations. Proportions result from many models.
6.5.22 In still water, a boat averages ______ mph. It takes the same amount of time to travel
_____ miles downstream, with the current as _______ miles upstream, against the current. What
is the rate of the water’s current?
For work problems involving multiple workers, such as people, copiers, pumps, etc., it is often
helpful to consider rate of work.
Rate of Work
The rate of work is the number of jobs that can be completed in a given unit of time. If one job
1
can be completed in t units of time, then the rate of work is given by .
t
We can add rates but not times. For example, if it takes April 4 hours to paint a room and Anisa
2 hours to paint the same room, it would not take 4 + 2 = 6 hours for them to paint the room
together. It cannot take longer for two people to do a job than it would take either one alone.
When dealing with two workers, we use the following formula:
1 1 1
  .
t1 t2 t
Here, t1 and t2 are the individual times to complete one job, and t is the time to complete the
job when working together.
6.5.31 A garden hose can fill a hot tub in _________ minutes, whereas the drain on a hot tub can
empty it in ________ minutes. If the drain is accidentally left open, how long will it take to fill
the tub?