Download Rational Expressions

Objective 6.1A
Vocabulary to Review
polynomial [5.2A]
evaluate a function [3.2A]
domain of a function [3.2A]
New Vocabulary
rational expression
rational function
Discuss the Concepts
Which of the following functions are rational expressions. Explain your answer.
1. f  x  
6
x 1
2. f  a  
a
a4
3. g  b  
1
b 2  1
4. h  c  
3
x2
5. C  x  
x5
3
4 and 5
Concept Check
1. Given f  x  
2x
, which is greater, f  2 or f  2  ?
x 1
2. What vertical line will the graph of f  x  
f(2)
2
never intersect?
x5
3. How many real numbers are excluded from the domain of f  x  
x=5
x2  3
?
4 x  2 x  1 x  7 
Three
Optional Student Activity
The cost C, in dollars, to remove p% of the salt in a tank of sea water is given by the rational
function C  p  
2000 p
, 0 ≤ p < 0.
100  p
a. Find the cost of removing 40% of the salt.
b. Find the cost of removing 80% of the salt.
$1333.33
$8000
c. Sketch a graph of C.
Objective 6.1B
New Vocabulary
simplest form of a rational expression
Properties to Review
Multiplication Property of One
[1.3A]
Discuss the Concepts
1. State the procedure for simplifying a rational expression.
2. Why can the numerator and denominator of a rational expression be divided by their common factors?
3. The denominator of a rational function is x2 + 5. Why are no real numbers excluded from the domain of
this function?
Concept Check
For each of the following, if the simplification is correct, write OK. If it is incorrect, rewrite the simplification
correctly.
1.
12 x 2  6 x 3 x (4 x  2)

 4x  2
3x
3x
2.
14 y 2  7 y 7 y (2 y  1)

 7(2 y  1)
y
y
3. 
3x  2 3 x  2

5
x
x
5b2  3b 5 b 2  3b

 3b  5
b2
b2
5.
4 x3  2 x 2 x (2 x 2  0)

 2 x2
2x
2x
1
1
1
1
1
1
OK
3x  2
x
4.
x 2  3 x  4 1 1 1 1


6. 2
x  3 x  4 111 3
OK
5b  3
b
2x2 + 1
 x  1 x  4
x 2  3x  4
Optional Student Activity
x4
for the following values of x: 1, 1.5, 1.75, 1.95, 1.999. Describe
x2
x4
the graph of the function f  x  
as x approaches 2 from the left.
x2
1. a. Evaluate the function f  x  
-5, -11, -23, -119, -5999; The graph decreases.
x4
for the following values of x: 3, 2.5, 2.25, 2.05, 2.001. Describe
x2
x4
the graph of the function f  x  
as x approaches 2 from the right.
x2
b. Evaluate the function f  x  
7, 13, 25, 121, 6001; The graph increases.
2x
for the following values of x: -4, -3.5, -3.2, -3.05, -3.001. Describe
x3
2x
the graph of the function f ( x) 
as x approaches -3 from the left.
x3
2. a. Evaluate the function f  x  
8, 14, 32, 122, 6002; The graph increases.
2x
for the following values of x: -2, -2.5, -2.8, -2.9, -2.999. Describe
x3
2x
the graph of the function f ( x) 
as x approaches -3 from the right.
x3
b. Evaluate the function f  x  
-4, -10, -28, -58, -5998; The graph decreases.
Objective 6.1C
Rules to Review
Product of two fractions:
a c ac
 
b d bd
[1.2B]
Discuss the Concepts
1. State the procedure for multiplying two rational expressions.
2. Is the procedure for multiplying rational expressions the same as that for multiplying arithmetic fractions?
Concept Check
Find two different pairs of rational expressions whose product is
x3
2x  1
2x  1
x3
and
, or
and
x4
3x  1
x4
3x  1
Optional Student Activity
 a 3  b 
 

 b   a 3
2
1. Simplify: 
3
b
a3
2x2  7 x  3
.
3x 2  13x  4
3
 x4  y 
2. Simplify:  2  

 y   x4
 x  2
n
 x  2
x4
y4
2
m
3. If
 x 2  4 x  4, what is the relationship between m and n?
m=n+2
Objective 6.1D
Vocabulary to Review
reciprocal
[1.2A]
New Vocabulary
reciprocal of a rational expression
Rules to Review
division of fractions:
a c a d ad
   
b d b c bc
[1.2B]
Discuss the Concepts
1. Write the reciprocal of each expression, and state the procedure for finding the reciprocal.
a.
x 2  x  30
x2  6 x  7
b.
3 y 2 z  12 yz
8 yz  32 z
c. 45ab2
2. State the procedure for dividing two rational expressions.
Concept Check
1. The product of a rational number and its reciprocal is 1. Show that this is true for the
rational expressions
2. Simplify:
y2 x y
 
x 2 x
3. Simplify:
ab a a
 
3 b2 4
5
2x2  4x  2
and
.
7
x  x  1
xy
2
4a
3b
Optional Student Activity
3
 2x   x 
  
 y   3y 
2
72x
y
1. Simplify: 
c
 3
2
c c
 
 2 4
2. Simplify:    
8
9
3. Complete the equation:
2x  6
3
 ? 
2
2
6 x  15 x
4 x  12
18 x 2  45 x
Answers to Writing Exercises
1. A rational function is a function that is written as an expression in which the numerator and denominator
are polynomials. An example is f  x  
x2  2x  3
.
7x  4
2. The domain of a rational function excludes all numbers for which the value of the polynomial in the
denominator is zero.
Answers to Writing Exercises
25. A rational expression is in simplest form when the numerator and denominator have no common factors
other than 1.
26. The expressions are not equal for all values of x. The expression
x  x  2
2  x  2
is undefined for x  2 , but
x
has the value 1 for x  2 .
2
Objective 6.2A
Vocabulary to Review
LCM of the denominators [1.2B]
New Vocabulary
least common multiple (LCM) of two or more polynomials
Discuss the Concepts
1. Why is the LCM of x  1 and x  2 the expression  x  1 x  2 ?
2. When is the LCM of two expressions equal to their product?
common factors
When the two expressions have no
Concept Check
If the statement is correct, write “Correct.” If the statement is not correct, explain the error.
1. Rewriting two fractions in terms of the LCM of their denominators is the reverse process of simplifying the
fractions. Correct
2. We can rewrite
2
x
4x
as
by using the Multiplication Property of One.
y
4y
5
3. The LCM of x , x , and x8 is x 2 .
Correct
Incorrect. The LCM is x8.
Optional Student Activity
What is the coefficient of the x3 term in the least common multiple of the polynomials 3x2  x  2 ,
3x2  8x  4 , and x3  2 x2  x  2 ?
-8
Objective 6.2B
Rules to Review
addition of fractions:
a b ab
 
c c
c
[1.2B]
subtraction of fractions:
a b a b
 
c c
c
[1.2B]
Discuss the Concepts
1. Why must rational expressions have the same denominator before they can be added or subtracted?
2. Explain the procedure for subtracting rational expressions with different denominators.
3. In adding and subtracting fractions, any common denominator will do. Explain the advantages and
disadvantages of using the LCM of the denominators.
Concept Check
1. One side of a triangle measures
8
9
15
in. The other two sides measure
in. and
in. Find the
3x
2x
6x
perimeter of the triangle.
29
in.
3x
2. Find the rational expression in simplest form that represents the sum of the reciprocals of the
consecutive integers x and x  1 .
3. Complete the equation:
2x  1
x  x  1
1
5
? 
2x  3
2x
8 x  15
2 x  2 x  3
4. -Complete the equation:
2b  1
b2  1
? 
b
b2  b
b2
b
Optional Student Activity
1. If
B
5 x  11
A

, find the value of

2 x  3 2 x2  x  6
x2
a. A
b. B
3
-1
1
2
1
2

and
by adding the numerators and the denominators:

5
3
5
3
1 2 3
1 2
3
 . Write the fractions , , and in order from smallest to largest. Now take any two other
53 8
5 3
8
2. A student incorrectly tried to add
fractions, add the numerators and the denominators, and write the fractions in order from smallest to
largest. If you see a pattern, explain it. If not, try a few more examples until you can find a pattern and
explain it.
Objective 6.3A
New Vocabulary
complex fraction
Discuss the Concepts
Solve Examples 1 and 2 of this objective by using the alternative method of rewriting the numerator and
denominator of the complex fraction as a single fraction and then dividing the numerator by the
denominator. Which method do you prefer? Why?
Concept Check
x
2
x
 5 and show that the
Have students find the reciprocal, in simplest form, of the complex fraction
x2
4 x 2  25
product of the complex fraction and its reciprocal is 1.
Note: The reciprocal is
x
.
2x  5
Optional Student Activity
According to the theory of relativity, the mass of a moving object is given by an equation that contains a
complex fraction. The equation is
m
m0
v2
1 2
c
, where m is the mass in grams of the moving object, m0 is the mass of the object at rest, v is
the speed of the object, and c is the speed of light.
1. Evaluate the expression at speeds of 0.5c, 0.75c, 0.90c, 0.95c, and 0.99c when the mass of the object at
rest is 10 g.
11.547, 15.119, 22.942, 32.026, 70.888
2. Explain how m changes as the speed of the object becomes closer to the speed of light.
As the velocity v of the object approaches the velocity of light c, the mass of the object increases.
3. Explain how this equation can be used to support the theory that an object cannot travel at the speed of
light.
According to the theory of relativity, the mass of an object at the speed of light would be infinite and would
require an infinite force to accelerate it. Because this is impossible, the theory suggests that an object of
any mass cannot attain the speed of light.
Answers to Writing Exercises
1. A complex fraction is a fraction whose numerator or denominator contains one or more fractions.
2. The general goal is to have no fractions in the numerator or denominator. The resulting fraction is then
written in simplest form.
Objective 6.4A
Vocabulary to Review
LCM of the denominators [1.2B]
New Vocabulary
ratio
rate
proportion
Discuss the Concepts
1. Provide two examples of proportions, one involving rates and one involving ratios.
2. Explain how to solve a proportion.
Concept Check
For what values of x are
x3
x6
and
equal?
x4
x8
x=0
Optional Student Activity
In 1950, the marriage rate per 1000 people in the United States was 11.1. The U.S. population in 1950 was
152,271,000. Find the number of marriages in the United States in 1950. Round to the nearest thousand.
1,690,000 marriages
Objective 6.4B
Concept Check
1. A cube of silver, 2 in. on each side, weighs 4 lb and is worth $280. How much is a 3-inch cube of silver
worth?
$945
2. Assume that it takes a man 2 h to dig a hole that is 2 m wide, 2 m long, and 2 m deep. Digging at the
same rate, how long would it take him to dig a hole 4 m wide, 4 m long, and 4 m deep?
16 h
Optional Student Activity
1. How many minutes does it take a clock’s hour hand to move through one degree of revolution?
2. If
aaa
 3 , what is the value of a 2 ?
aaa
2 min
9
3. A team won 40 games out of 60 played. How many more games must the team win in succession to
raise its record of wins to 80%?
40 games
4. A number h is the harmonic mean of the numbers a and b if the reciprocal of h is equal to the average of
the reciprocals of a and b.
a. Write an expression for the harmonic mean of a and b.
b. Find the harmonic mean of 10 and 15.
2ab
ab
12
Answers to Writing Exercises
1. A ratio is the quotient of two quantities that have the same unit. A rate is the quotient of two quantities
that have different units.
2. A proportion is an equation that states that two ratios or rates are equal.
Answers to Writing Exercises
35.
a
c

b
d
Add 1 to each side.
a
c
1 
1
b
d
Rewrite each side of the equation as a single fraction.
a
b
c
d

 
b
b d
d
ab
cd

b
d
Objective 6.5A
Vocabulary to Review
LCM of the denominators [1.2B]
quadratic equation [5.7A]
literal equation [2.1D]
New Vocabulary
clearing denominators
Discuss the Concepts
1. What is a rational equation?
2. Explain why it is necessary to check the solution of a rational equation.
3. Determine whether each statement is always true, sometimes true, or never true.
a. The first step in solving an equation containing fractions is to find the LCM of the denominators.
Always true
b. The process of clearing denominators in an equation containing fractions is an application of the
Multiplication Property of Equations.
Always true
c. Both sides of an equation containing fractions can be multiplied by the same number without changing
the solution of the equation.
Sometimes true
Concept Check
1. Are the solutions of the equations shown below the same? Explain your answer.
2x  9x
1
1

2x 9x
No. The solution to the first equation is 0, and the second equation has no solution.
2. The sum of a number and its reciprocal is
25
. Find the number.
12
3. The sum of the multiplicative inverses of two consecutive integers is
3
4
or
3
4
11
. Find the integers.
30
Optional Student Activity
Write an original rational equation that has
a. 0 as its only solution.
b. no solution.
c. one solution that does not check and a second solution that does check.
Objective 6.5B
5 and 6
New Vocabulary
rate of work
New Formulas
Rate of work  time worked  part of task completed
Discuss the Concepts
1. It takes a janitorial crew 5 h to clean a company’s offices. What fraction of the job does the crew
complete in x hours?
x
of the job
5
2. Only two people worked on a job, and together they completed it. One person completed
t
of the job,
30
t
of the job. Write an equation to express the fact that together they
20
t
t

1
30 20
and the other person completed
completed the whole job.
Concept Check
A painter can paint a ceiling in 60 min. The painter’s apprentice can paint the same ceiling in 90 min.
a. How long will it take to paint the ceiling if they work together?
36 min
b. What fraction of the job does the painter complete? What fraction of the job does the apprentice
complete? Are these fractions equal to the complete job?
Painter:
3
2
3 2
; apprentice: ; Yes.   1 , the complete job
5
5
5 5
c. Could the answer to part (a) ever be more than 60 min? Why or why not?
No. The painter can complete the entire job in 60 min. With the apprentice helping, it must take less time
to do the job than it does for the painter working alone.
Optional Student Activity
1. A bricklayer is installing a walkway in front of a house. She figures it will take her 6 days to complete the
job. However, on the fifth day, the temperature rises and she works more slowly, completing only
1
of
12
the job. Her rate of work is back to normal the next day. How long does it take her to complete the entire
job?
6
1
days
2
2. One press can print the weekly edition of a newspaper in 12 h, a second press can complete the job in 8
h, and a third press can complete the job in 6 h. How long would it take to print the newspaper with all
three presses operating?
2
Objective 6.5C
Vocabulary to Review
uniform motion
[2.3C]
New Formulas
Distance  rate  time
or
2
h
3
Distance ÷ rate  time
Discuss the Concepts
1. A plane flies 300 mph in calm air and the rate of the wind is r mph.
a. Write an expression to represent the rate of the plane flying with the wind.
300 + r
b. Write an expression to represent the rate of the plane flying against the wind. 300 – r
2. The rate of a river’s current is 2 mph. The rowing rate of a rowing crew in calm water is r.
a. Write an expression to represent the rate of the rowing crew when traveling with the current.
r+2
b. Write an expression to represent the rate of the rowing crew when traveling against the current. r – 2
Concept Check
Marlys can row a boat 3 mph faster than she can swim. She is able to row 10 mi in the same time it takes
her to swim 4 mi. Find the rate at which Marlys swims. 2 mph
Optional Student Activity
1. A plane flew from St. Louis to Boston, a distance of d miles, at an average rate of 400 mph. Because of
prevailing winds, on the return trip the plane flew at an average rate of 500 mph.
a. Write an expression for the total flying time.
b. Find the average rate for the entire round trip.
9d
2000
444.4 mph
2. For the first 5 mi of a 10-mile race, Ray’s rate was 10 mph. For the last 5 mi, his rate slowed to 8 mph.
How long did it take Ray to complete the race? What was his average rate for the race?
1
1
8
h; 8 mph
8
9
3. Friends who live 40 mi apart decide to bicycle toward each other and meet for a picnic lunch. The first
friend bikes at a rate of 20 mph, and the second friend bikes at a rate of 15 mph. How far from the first
friend’s home will they meet for lunch?
22
6
mi
7
4. By increasing your speed by 5 mph, you can drive the 165-mile trip to your hometown in 15 min less time
than it usually takes you to drive the trip. How fast do you usually drive?
55 mph
Answers to Writing Exercises
56. In simplifying a rational expression, the denominator is factored; any factors common to the numerator
and denominator are eliminated. In solving a rational equation, we clear the denominators by
multiplying each side of the equation by the LCM of the denominators; the result is an equation that
does not contain any fractions.
Objective 6.6A
Vocabulary to Review
constant [1.3B]
New Vocabulary
direct variation
constant of variation
constant of proportionality
inverse variation
joint variation
combined variation
Discuss the Concepts
1. What is the difference between a direct variation and an inverse variation?
2. Given that x and y vary directly, how can the constant of variation be defined?
k
y
x
3. Provide examples of two quantities that vary directly and examples of two quantities that vary inversely.
Here are two examples that might interest students:
(1) The radius of a hailstone varies directly as the time that the hailstone spent in a high cloud.
(2) The wing flapping rate, in beats per second, of a bird is inversely proportional to the length of its
wings.
Concept Check
Determine whether the following statements are true or false.
1. If x varies inversely as y, then when x is doubled, y is doubled.
2. If a varies inversely as b, then ab is a constant.
3. If a varies jointly as b and c, then a 
kb
.
c
False
True
False
4. If the length of a rectangle is held constant, then the area of the rectangle varies directly as the width.
True
5. If the area of a rectangle is held constant, then the length varies directly as the width.
False
6. The circumference of a circle varies directly as the diameter. If the diameter of a circle is doubled, then
the circumference of the circle will be doubled. True
Optional Student Activity
The number of times a wheel rotates in traveling one mile is inversely proportional to the diameter of the
wheel. A bicycle tire with a 27-inch diameter makes 747 revolutions in traveling one mile. Find the diameter
of a bicycle tire that would travel one mile in only 100 revolutions. Round to the nearest whole number.
202 in.
Optional Student Activity
Variation equations are introduced as functions. You might ask students to compare and contrast the
graphs of the direct variation equations shown on page 377 with those of the inverse variation equations
shown on page 378, noting, for example, the behavior of the dependent variable as the independent
variable changes. Students should note that the domain of the application in the second HOW TO on page
377 is {x| x > 0} and the domain of the application in the second HOW TO on page 378 is {w| w > 0}.
Answers to Focus on Problem Solving: Implication
1. Contrapositive: If I do not live in Illinois, then I do not live in Chicago.
Converse:If I live in Illinois, then I live in Chicago.
2. Contrapositive: If yesterday was not May 31, then today is not June 1.
Converse: If yesterday was May 31, then today is June 1. Today is June 1 if and only if yesterday was
May 31.
3. Contrapositive: If tomorrow is Friday, then today is Thursday.
Converse: If tomorrow is not Friday, then today is not Thursday.
Today is not Thursday if and only if tomorrow is not Friday.
4. Contrapositive: If a number is not divisible by 4, then it is not divisible by 8.
Converse: If a number is divisible by 4, then it is divisible by 8.
5. Contrapositive: If a number is not divisible by 2, then it is not an even number.
Converse: If a number is divisible by 2, then it is an even number.
A number is an even number if and only if it is divisible by 2.
6. Contrapositive: If a number is not a multiple of 3, then it is not a multiple of 6.
Converse: If a number is a multiple of 3, then it is a multiple of 6.
7. Contrapositive: If z  5 , then 4 z  20 .
Converse: If z  5 , then 4 z  20 .
4 z  20 if and only if z  5 .
8. Contrapositive: If an angle is not a right angle, then it does not measure 90°.
Converse: If an angle is a right angle, then it measures 90°.
An angle measures 90° if and only if it is a right angle.
9. Contrapositive: If p is not an odd number, then p is not a prime number greater than 2.
Converse: If p is an odd number, then p is a prime number greater than 2.
10. Contrapositive: If the graph of an equation is not a straight line, then the equation of the graph is not of
the form y  mx  b .
Converse:If the graph of an equation is a straight line, then the equation of the graph is y  mx  b .
The equation of a graph is y  mx  b if and only if the graph of the equation is a straight line.
11. Contrapositive: If ab  0 then a  0 or b  0 .
Converse: If ab  0, then a  0 or b  0 .
a  0 or b  0 if and only if ab  0 .
12. Contrapositive: If a point is not on the x-axis, then its coordinates are not  5,0  .
Converse: If a point is on the x-axis, then the coordinates of the point are  5,0  .
13. Contrapositive: If a quadrilateral does not have four sides of equal length, then the quadrilateral is not a
square.
Converse:If a quadrilateral has four sides of equal length, then the quadrilateral is a square.
2
2
14. Contrapositive: If x  y , then x  y .
2
2
Converse: If x  y , then x  y .
Answers to Projects and Group Activities: Graphing Variation Equations
1. The graph represents a linear function.
2. The graph represents a linear function.
3. The graph is the graph of a function.
Answers to Projects and Group Activities: Transformers
1. 80 amperes
2. 300 amperes
3. 160 amperes
4. 36 volts
5. 0.9 volts
6. 10 volts