• rational equation
• weighted average
• rational inequality
Solve a Rational Equation
Solve
. Check your solution.
The LCD for the terms is 24(3 – x).
Original equation
Multiply each side
by 24(3 – x).
Solve a Rational Equation
Distributive Property
Simplify.
Simplify.
Add 6x and –63 to
each side.
Solve a Rational Equation
Original equation
Check
x = –45
Simplify.
Simplify.

The solution is
correct.
Solve a Rational Equation
Answer: The solution is –45.
Solve
.
A. –2
B.
0%
B
A
0%
A
B
C
0%
D
D
D. 2
C
C.
A.
B.
C.
0%
D.
Solve a Rational Equation
Solve
Check your solution.
The LCD is (p + 1)(p – 1).
Original equation
Multiply by the
LCD.
Solve a Rational Equation
(p – 1)(p2 – p – 5) = (p2 – 7)(p + 1) + p(p + 1)(p – 1)
Divide common
factors.
p3 – p2 – 5p – p2 + p + 5 = p3 + p2 – 7p – 7 + p3 – p
Distributive
Property
p3 – 2p2 – 4p + 5 = 2p3 + p2 – 8p – 7
Simplify.
0 = p3 + 3p2 – 4p – 12
Subtract
p3 – 2p2 – 4p + 5
from each side.
Solve a Rational Equation
0 = (p + 3)(p + 2)(p – 2)
Factor.
0 = p + 3 or 0 = p + 2 or 0 = p – 2
Zero Product
Property
Check
Try p = –3.
Original equation
p = –3
Solve a Rational Equation
Simplify.
Simplify.

Try p = –2.
Original equation
Solve a Rational Equation
p = –2
Simplify.
Simplify.

Solve a Rational Equation
Try p = 2.
Original equation
p=2
Simplify.
Simplify.

Answer: The solutions are
–3, –2 and 2.
A. 4
A
0%
0%
B
D. –4
A
B
C
0%
D
D
C. 2
A.
B.
C.
0%
D.
C
B. –2
Mixture Problem
BRINE Aaron adds an 80% brine (salt and water)
solution to 16 ounces of solution that is 10% brine.
How much of the solution should be added to
create a solution that is 50% brine?
Understand
Aaron needs to know how much of a
solution needs to be added to an
original solution to create a new
solution.
Mixture Problem
Plan
Each solution has a certain
percentage that is salt. The
percentage of brine in the final
solution must equal the amount of
brine divided by the total solution.
Percentage of brine in solution
Mixture Problem
Solve
Write a proportion.
Substitute.
Simplify numerator.
LCD is 100(10 + x).
Mixture Problem
Divide common
factors.
Simplify.
Distribute.
Subtract 50x and
160.
Divide each side
by 30.
Answer: Aaron needs to add
brine solution.
ounces of 80%
Mixture Problem
Check
Original
equation
?
?
0.5 = 0.5 
Simplify.
Simplify.
BRINE Janna adds a 65% base solution to
13 ounces of solution that is 20% base. How much
of the solution should be added to create a solution
that is 40% base?
A. 9.6 ounces
0%
B
D. 12.3 ounces
A
0%
A
B
C
0%
D
D
C. 11.8 ounces
C
B. 10.4 ounces
A.
B.
C.
0%
D.
Distance Problem
SWIMMING Lilia swims for 5 hours in a stream that
has a current of 1 mile per hour. She leaves her dock
and swims upstream for 2 miles and then back to her
dock. What is her swimming speed in still water?
Understand
We are given the speed of the current,
the distance she swims upstream, and
the total time.
Plan
She swam 2 miles upstream against the
current and 2 miles back to the dock with
the current. The formula that relates
distance, time, and rate is d = rt or
Distance Problem
Let r equal her speed in still water. Then
her speed with the current is r + 1, and
her speed against the current is r – 1.
Time going with
the current
plus
time going against
the current
total
equals time.
5
Solve
Original equation
Distance Problem
Multiply each
side by r2 – 1.
Divide
Common
Factors
(r + 1)2 + (r – 1)2 = 5(r2 – 1)
Simplify.
Distribute.
Simplify.
Subtract 4r
from each side.
Distance Problem
Use the Quadratic Formula to solve for r.
Quadratic Formula
x = r, a = 5, b = – 4,
and c = –5
Simplify.
Simplify.
Distance Problem
r ≈ 1.5 or –0.7
Use a calculator.
Answer: Since speed must be positive, the answer is
about 1.5 miles per hour.
Original equation
Check
?
r = 1.5
?
Simplify.

Simplify.
SWIMMING Lynne swims for 1 hour in a stream
that has a current of 2 miles per hour. She leaves
her dock and swims upstream for 3 miles and then
back to her dock. What is her swimming speed in
still water?
C. about 4.6 mph
D. about 6.6 mph
0%
B
A
0%
A
B
C
0%
D
D
B. about 2.0 mph
A.
B.
C.
0%
D.
C
A. about 0.6 mph
Work Problems
MOWING LAWNS Wuyi and Uima mow lawns
together. Wuyi working alone could complete a
particular job in 4.5 hours, and Uima could
complete it alone in 3.7 hours. How long does it
take to complete the job when they work together?
Understand
Plan
We are given how long it takes Wuyi
and Uima working alone to mow a
particular lawn. We need to determine
how long it would take them together.
Wuyi can mow the lawn in 4.5 hours,
so the rate of mowing is
lawn per hour.
of a
Work Problems
Uima can mow the lawn in 3.7 hours,
so the rate of mowing is
per hour.
The combined rate is
of a lawn
Work Problems
Solve
Write the equation.
Add
Multiply both sides
by x.
x ≈ 2.0304
Multiply 1 by
Answer: It would take Wuyi and Uima about 2 hours
to mow the lawn together.
Work Problems
Check
Original
equation
?
x≈2

Simplify.
PAINTING Adriana and Monique paint rooms
together. Adriana working alone could complete a
particular job in 6.4 hours, and Monique could
complete it alone in 4.8 hours. How long does it
take to complete the job when they work together?
C. about 2 hours and 45 minutes
0%
B
D. about 2 hours and 56 minutes
A
0%
A
B
C
0%
D
D
B. about 2 hours and 36 minutes
A.
B.
C.
0%
D.
C
A. about 2 hours and 28 minutes
Solve a Rational Inequality
Solve
Step 1
Values that make the denominator equal to
0 are excluded from the denominator. For this
inequality the excluded value is 0.
Step 2
Solve the related equation.
Related equation
Solve a Rational Inequality
Multiply each side by 9k.
Simplify.
Add.
Divide each side by 6.
Solve a Rational Inequality
Step 3
Draw vertical lines at the excluded value and
at the solution to separate the number line
into regions.
Now test a sample value in each region to
determine if the values in the region satisfy
the inequality.
Solve a Rational Inequality
Test k = –1.

k < 0 is a solution.
Solve a Rational Inequality
Test k =
.

0<k<
is not a solution.
Solve a Rational Inequality
Test k = 1.

Solve
.
A. x < 0
B. x > 0
0%
B
A
0%
A
B
C
0%
D
D
D. 0 < x < 4
C
C. x < 0 or x > 4
A.
B.
C.
0%
D.