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CHAPTER
CHAPTER
TABLE OF CONTENTS
2-1 Rational Numbers
2-2 Simplifying Rational
Expressions
2-3 Multiplying and Dividing
Rational Expressions
2-4 Adding and Subtracting
Rational Expressions
2-5 Ratio and Proportion
2-6 Complex Rational
Expressions
2-7 Solving Rational Equations
2-8 Solving Rational Inequalities
Chapter Summary
Vocabulary
Review Exercises
Cumulative Review
THE
RATIONAL
NUMBERS
When a divided by b is not an integer, the quotient
is a fraction.The Babylonians, who used a number sys-
tem based on 60, expressed the quotients:
20 8 as instead of
21 8 as instead of
Note that this is similar to saying that 20 hours
divided by 8 is 2 hours, 30 minutes and that 21 hours
divided by 5 is 2 hours, 37 minutes, 30 seconds.
This notation was also used by Leonardo of Pisa
(1175–1250), also known as Fibonacci.
The base-ten number system used throughout the
world today comes from both Hindu and Arabic math-
ematicians. One of the earliest applications of the
base-ten system to fractions was given by Simon Stevin
(1548–1620), who introduced to 16th-century Europe
a method of writing decimal fractions. The decimal
that we write as 3.147 was written by Stevin as
3 ^1 ^4 ^7 ^ or as 3
1
4
7
. John Napier
(1550–1617) later brought the decimal point into com-
mon usage.
2
2 1
1
2
2 1
When persons travel to another country, one of the first things that they learn is
the monetary system. In the United States, the dollar is the basic unit, but
most purchases require the use of a fractional part of a dollar. We know
that a penny is $0.01 or of a dollar, that a nickel is $0.05 or of a
dollar, and a dime is $0.10 or of a dollar. Fractions are common in our
everyday life as a part of a dollar when we make a purchase, as a part of a pound
when we purchase a cut of meat, or as a part of a cup of flour when we are
baking.
In our study of mathematics, we have worked with numbers that are not
integers. For example, 15 minutes is or of an hour, 8 inches is or of a foot,
and 8 ounces is or of a pound. These fractions are numbers in the set of
rational numbers.
For every rational number that is not equal to zero, there is a multiplicative
inverse or reciprocal such that 5 1. Note that. If the non-zero
numerator of a fraction is equal to the denominator, then the fraction is equal
to 1.
EXAMPLE 1
Write the multiplicative inverse of each of the following rational numbers:
Answers
a.
b.
c. 5
Note that in b , the reciprocal of a negative number is a negative number.
a
b?^
b
a 5
ab ab
a
b?^
b a
b a
a b
2-1 RATIONAL NUMBERS
40 The Rational Numbers
DEFINITION
A rational number is a number of the form where a and b are integers and
b 0.
a b
EXAMPLE 2
Find the common fractional equivalent of.
Solution Let x 5 5 0....
How to Proceed
(1) Multiply the value of x by 100 to write a
number in which the decimal point follows
the first pair of repeating digits:
(2) Subtract the value of x from both sides of
this equation:
(3) Solve the resulting equation for x and simplify
the fraction:
Check The solution can be checked on a calculator.
ENTER : 2 11
DISPLAY:
Answer
EXAMPLE 3
Express as a common fraction.
Solution: Let x 5 0.12484848...
How to Proceed
(1) Multiply the value of x by the power of 10 that
makes the decimal point follow the first set of
repeating digits. Since we want to move the
decimal point 4 places, multiply by 10 4 5 10,000:
(2) Multiply the value of x by the power of 10 that
makes the decimal point follow the digits that
do not repeat. Since we want to move the
decimal point 2 places, multiply by 10 2 5 100:
(3) Subtract the equation in step 2 from the
equation in step 1:
(4) Solve for x and reduce the fraction to
lowest terms:
Answer
ENTER
42 The Rational Numbers
100 x 5 18....
x 5
99x 5 18
2 x 5 20.181818c
100x 5 18.181818c
10,000 x 5 1,248.4848...
100 x 5 12.4848...
x 5
9,900x 5 1,
2 100x 5 2 12.4848c
10,000x 5 1,248.4848c
Writing About Mathematics
1. a. Why is a coin that is worth 25 cents called a quarter?
b. Why is the number of minutes in a quarter of an hour different from the number of
cents in a quarter of a dollar?
2. Explain the difference between the additive inverse and the multiplicative inverse.
Developing Skills
In 3–7, write the reciprocal (multiplicative inverse) of each given number.
In 8–12, write each rational number as a repeating decimal.
In 13–22, write each decimal as a common fraction.
Exercises
Rational Numbers 43
Procedure
To convert an infinitely repeating decimal to a common fraction:
1. Write the equation: x 5 decimal value.
2. Multiply both sides of the equation in step 1 by 10 m , where m is the number
of places to the right of the decimal point following the first set of repeating
digits.
3. Multiply both sides of the equation in step 1 by 10 n , where n is the number
of places to the right of the decimal point following the non -repeating digits.
(If there are no non-repeating digits, then let n 5 0.)
4. Subtract the equation in step 3 from the equation in step 2.
5. Solve the resulting equation for x , and simplify the fraction completely.
EXAMPLE 2
Simplify: Answers
a.
b.
c.
Note: We must eliminate any value of the variable or variables for which the
denominator of the given rational expression is zero.
The rational expressions , , and in the example shown above are
in simplest form because there is no factor of the numerator that is also a factor
of the denominator except 1 and 2 1. We say that the fractions have been
reduced to lowest terms.
When the numerator or denominator of a rational expression is a mono-
mial, each number or variable is a factor of the monomial. When the numerator
or denominator of a rational expression is a polynomial with more than one
term, we must factor the polynomial. Once both the numerator and denomina-
tor of the fraction are factored, we can reduce the fraction by identifying factors
in the numerator that are also factors in the denominator.
In the example given above, we wrote:
We can simplify this process by canceling the common factor in the numer-
ator and denominator.
( y 2, 3)
Note that canceling ( y 2 2) in the numerator and denominator of the frac-
tion given above is the equivalent of dividing ( y 2 2) by ( y 2 2). When any num-
ber or algebraic expression that is not equal to 0 is divided by itself, the quotient
is 1.
(y 2 2 )
1
(y 2 2 ) 1
(y 2 3 )
y 2 3
y 2 2 y^2 2 5y 1 6
y 2 2
(y 2 2 )(y 2 3 ) 5
y 2 2
y 2 2?^
y 2 3 5 1?^
y 2 3 5
y 2 3 (y^2 2,^3 )
y 2 3
3b a 2 1
x 3
y 2 2 (y 2 2 )(y 2 3 )
y 2 2 y 2 2
y 2 3
y 2 3
y 2 3
(y 2 2, 3 )
y 2 2 y^2 2 5y 1 6
a a
3b a 2 1
3b a 2 1
3b a 2 1
(a 2 0, 1)
3ab a(a 2 1 )
x 3
x 3
x 3
2x 6
Simplifying Rational Expressions 45
EXAMPLE 3
Write in lowest terms.
Solution METHOD 1 METHOD 2
Answer ( x 0)
Factors That Are Opposites
The binomials ( a 2 2) and (2 2 a ) are opposites or additive inverses. If we
change the order of the terms in the binomial (2 2 a ), we can write:
(2 2 a ) 5 ( 2 a 1 2) 5 21( a 2 2)
x 2 4 x
x 2 4 x
x 2 4 x
x 2 4
5 x
x 2 4 x
3x 2 12
3x 5
1 (x 2 4 ) 3 1
x
3x 2 12
3x 5
3 (x 2 4 ) 3x
3x 2 12 3x
46 The Rational Numbers
Procedure
To reduce a fraction to lowest terms:
METHOD 1
1. Factor completely both the numerator and the denominator.
2. Determine the greatest common factor of the numerator and the
denominator.
3. Express the given fraction as the product of two fractions, one of which has
as its numerator and its denominator the greatest common factor deter-
mined in step 2.
4. Write the fraction whose numerator and denominator are the greatest
common factor as 1 and use the multiplication property of 1.
METHOD 2
1. Factor both the numerator and the denominator.
2. Divide both the numerator and the denominator by their greatest common
factor by canceling the common factor.
In 11–30, write each rational expression in simplest form and list the values of the variables for
which the fraction is undefined.
Multiplying Rational Expressions
We know that and that. In general, the product of two
rational numbers and is for b 0 and d 0.
This same rule holds for the product of two rational expressions:
The product of two rational expressions is a fraction whose numerator is the
product of the given numerators and whose denominator is the product of
the given denominators.
For example:
( a 0)
This product can be reduced to lowest terms by dividing numerator and denom-
inator by the common factor, 4.
We could have canceled the factor 4 before we multiplied, as shown below.
Note that a is not a common factor of the numerator and denominator because
it is one term of the factor ( a 1 5), not a factor of the numerator.
3 (a 1 5 )
4a 3
a 5
3 (a 1 5 ) 4 1
a 3
3
a 5
9 (a 1 5 ) a^2
3 (a 1 5 )
4a?^
a 5
36 (a 1 5 ) 4a^2
9 (a 1 5 ) 4 1
a^2
9 (a 1 5 ) a^2
3 (a 1 5 )
4a?^
a 5
36 (a 1 5 ) 4a^2
a b
c d
ac bd
c d
a b
2-3 MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS
5 ( 1 2 b) 1 15 b 2 2 16
4 2 2 (x 2 1 ) x^2 2 6x 1 9
3 2 (b 1 1 ) 4 2 b^2
a^3 2 a^2 2 a 1 1 a^2 2 2a 1 1
27 1 7a 21a^2 2
5y^2 2 y^2 1 4y 1 4
x^2 2 7x 1 12 x^2 1 2x 2 15
4a^2 2 4a 1 8
2a 1 10 3a 1 15
3xy 9xy 1 6x^2 y^3
8c^2 8c^2 1 16c
10d 15d 2 20d^2
8ab 2 4b 2 6ab
9y 2 1 3y 6y^2
8a 1 16 12a
9cd^2 12c^4 d^2
14b^4 21b^3
12xy 2 3x^2 y
5a^2 b 10a
48 The Rational Numbers
EXAMPLE 1
a. Find the product of in simplest form.
b. For what values of the variable are the given fractions and the product
undefined?
Solution a. METHOD 1
How to Proceed
(1) Multiply the numerators of
the fractions and the
denominators of the
fractions:
(2) Factor the numerator and
the denominator. Note that
the factors of (5 b 1 15) are
5( b 1 3) and the factors of
(3 b^2 1 9 b ) are 3 b ( b 1 3).
Reduce the resulting
fraction to lowest terms:
12b 2 5b 1 15
b 2 2 9 3b
2 1 9b
Multiplying and Dividing Rational Expressions 49
Procedure
To multiply fractions:
METHOD 1
1. Multiply the numerators of the given fractions and the denominators of the
given fractions.
2. Reduce the resulting fraction, if possible, to lowest terms.
METHOD 2
1. Factor any polynomial that is not a monomial.
2. Cancel any factors that are common to a numerator and a
denominator.
3. Multiply the resulting numerators and the resulting denominators to write
the product in lowest terms.
5 Answer
4b(b 2 3 ) 5 (b 1 3 )
4b(b 2 3 ) 5 (b 1 3 )
3b(b 1 3 ) 3b(b 1 3 )
4b(b 2 3 ) 5 (b 1 3 )
12b 2 (b 1 3 )(b 2 3 ) 15b(b 1 3 )(b 1 3 )
12b^2 5b 1 15
b^2 2 3b^2 1 9b
12b^2 (b^2 2 9 ) (5b 1 15 )(3b^2 1 9b)
EXAMPLE 2
Divide and simplify:
Solution How to Proceed
(1) Use the reciprocal of
the divisor to write
the division as a
multiplication:
(2) Factor each polynomial:
(3) Cancel any factors that
are common to the
numerator and
denominator:
(4) Multiply the remaining
factors:
Answer ( a 0, 5)
EXAMPLE 3
Perform the indicated operations and write the answer in simplest form:
Solution Recall that multiplications and divisions are performed in the order in which
they occur from left to right.
5 Answer
2a
5 (a^2 2 3, 0)
2
5?^
a 3 1
3a
1
a 1 3 1
2 (a 1 3 )
1
5a 1
a 3
3a a 1 3
5a 2a 1 6
a
3a a 1 3
2 (a 1 3 ) 5a
a 3
3a
a 1 3 4
5a
2a 1 6?^
a
3 (a 1 2 ) a^2
a^2 2 3a 2 10 5a
a^2 2 5a 15
Multiplying and Dividing Rational Expressions 51
3 (a 1 2 )
a
2
(a 2 5 )
1 (a 1 2 ) 5 1
a?^
3
a(a 2 5 ) 1
(a 2 5 )(a 1 2 )
5a?^
a(a 2 5 )
a^2 2 3a 2 10
5a 4
a^2 2 5a
a^2 2 3a 2 10
5a?^
a^2 2 5a
Writing About Mathematics
1. Joshua wanted to write this division in simplest form:. He began by cancel-
ing ( x 2 2) in the numerator and denominator and wrote following:
Is Joshua’s answer correct? Justify your answer.
2. Gabriel wrote. Is Gabriel’s solution correct? Justify
your answer.
Developing Skills
In 3–12, multiply and express each product in simplest form. In each case, list any values of the vari-
ables for which the fractions are not defined.
In 13–24, divide and express each quotient in simplest form. In each case, list any values of the vari-
ables for which the fractions are not defined.
23. 24. (a
2
2a 1 2
(2x 1 7 ) 4 a
2x^2 1 5x 2 7
a^2 1 8a 1 15
4a 4 (a^1 3 )
4b 1 12
b 4 (b^1 3 )
w^2 2 w
5 w^4
w^2 2 5
c^2 2 6c 1 9
5c 2 15 4
c 2 3 5
6y^2 2 3y 3y
4y^2 2 2
x 2 2 3x
4x 2 8 9
a^2
8a 4
3a 4
6b 5c
3b 10c
a 4
4a
6 2 2x x^2 2
15 1 5x 4x
2a 1 4 6a
3a^2 a^2 1 2a
a^2 2 5a 1 4 3a 1 6
2a 1 4 a^2 2
7y 1 21 7y
y^2 2
a^2 2
3a?^
a^2 2a 2 20
b 1 1
4?^
5b 1 5
3a 5
9a
4y 5x
x 8y
7a
3a 20
12x 5x 1 10
12x 4 4 (5x 1 10 ) 4 5
3x x 1 2
x 2 2 1
4 (x 2 2 )
1
x 2 2 4
4 (x 2 2 ) 7
Exercises
52 The Rational Numbers
To add the fractions and , we need to find a denominator that is a mul-
tiple of both 2 x and of y. One possibility is their product, 2 xy. Multiply each frac-
tion by a fraction equal to 1 so that the denominator of each fraction will be 2 xy :
and
( x 0, y 0)
The least common denominator ( LCD ) is often smaller than the product
of the two denominators. It is the least common multiple ( LCM ) of the
denominators, that is, the product of all factors of one or both of the
denominators.
For example, to add , first find the factors of each denomina-
tor. The least common denominator is the product of all of the factors of the first
denominator times all factors of the second that are not factors of the first. Then
multiply each fraction by a fraction equal to 1 so that the denominator of each
fraction will be equal to the LCD.
Factors of 2 a 1 2: 2 ( a 1 1)
Factors of a^2 2 1: ( a 1 1) ( a 2 1)
LCD: 2 ( a 1 1) ( a 2 1)
and
Since this sum has a common factor in the numerator and denominator, it
can be reduced to lowest terms.
( a 2 1, a 1)
Any polynomial can be written as a rational expression with a denominator
of 1. To add a polynomial to a rational expression, write the polynomial as an
equivalent rational expression.
For example, to write the sum b 1 3 1 as a single fraction, multiply
( b 1 3) by 1 in the form.
2b 2 1 6b 1 1
2b (b^2 0 )
2b 2 1 6b 2b
2b
(b 1 3 ) 1
2b
b 1 3 1 A^
2b 2b B^
2b
2b 2b
2b
a 1 1 2 (a 1 1 )(a 2 1 )
a 1 1
1
2 (a 1 1 )(a 2 1 ) 1
2 (a 2 1 )
2a 1 2
a^2 2
a 2 1 1 2 2 (a 1 1 )(a 2 1 )
a 1 1 2 (a 1 1 )(a 2 1 )
2 (a 1 1 )(a 2 1 )
a 2 1 2 (a 1 1 )(a 2 1 )
a^2 2
(a 1 1 )(a 2 1 )
2a 1 2 5
2 (a 1 1 )
a 2 1 a 2 1
2a 1 2 1
a^2 2
2x 1
y 5
5y
2xy 1
6x
2xy 5
5y 1 6x 2xy
y 5
y?^
2x
2x 5
6x 2xy
2x 5
2x?^
y
y 5
5y 2xy
y
2x
54 The Rational Numbers
EXAMPLE 1
Write the difference as a single fraction in lowest terms.
Solution How to Proceed
(1) Find the LCD of the
fractions:
(2) Write each fraction as an
equivalent fraction with a
denominator equal to the
LCD:
(3) Subtract:
(4) Simplify:
(5) Reduce to lowest terms:
Answer ( x 2 3, 1, 3)
EXAMPLE 2
Simplify:
Solution STEP 1. Rewrite each expression in parentheses as a single fraction.
and
STEP 2. Multiply.
5 x 1 1
x 1 1 1
(x 1 1 )(x 2 1 )
1
x 1
x
1
x 2 1 1
A
x^2 2 x B A^
x x 2 1 B^
5 Q
(x 1 1 )(x 2 1 ) x
R A
x x 2 1 B
x x 2 1
x 2 1 1 1
5 x 2 1
x^2 2 x
x 2 1 x 2 1
x 2 1
x^2 x
x
x 2 1 5 1 A^
x 2 1 x 2 1 B^1
x 2 x 2 1
x 5 x^ A^
x x B^2
x
A x^2
x B A^
x 2 1 B
(x 2 3 )(x 1 3 )
x x^2 2 4x 1 3
x x^2 1 2x 2 3
Adding and Subtracting Rational Expressions 55
x^2 2 4 x 1 3 5 ( x 2 3) ( x 2 1)
x^2 1 2 x 2 3 5 ( x 2 1) ( x 1 3)
LCD 5 ( x 2 3) ( x 2 1) ( x 1 3)
(x 2 3 )(x 1 3 )
9x 2 9 (x 2 3 )(x 2 1 )(x 1 3 )
x^2 1 3x 2 (x^2 2 6x 1 9 ) (x 2 3 )(x 2 1 )(x 1 3 )
x x^2 2 4x 1 3
x 2 3 x^2 1 2x 2 3
x^2 2 6x 1 9 (x 2 3 )(x 2 1 )(x 1 3 )
x 2 3 x^2 1 2x 2 3
x 2 3 (x 1 3 )(x 2 1 )
x 2 3 x 2 3
x^2 1 3x (x 2 3 )(x 2 1 )(x 1 3 )
x x^2 2 4x 1 3
x (x 2 3 )(x 2 1 )
x 1 3 x 1 3
Applying Skills
In 21–24, the length and width of a rectangle are expressed in terms of a variable.
a. Express each perimeter in terms of the variable.
b. Express each area in terms of the variable.
21. l 5 2 x and w 5
22. l 5 3 x 1 3 and w 5
23. l 5 and w 5
24. l 5 and w 5
We often want to compare two quantities that use the same unit. For example,
in a given class of 25 students, there are 11 students who are boys. We can say
that of the students are boys or that the ratio of students who are boys to all
students in the class is 11 : 25.
A ratio, like a fraction, can be simplified by dividing each term by the same
non-zero number. A ratio is in simplest form when the terms of the ratio are
integers that have no common factor other than 1.
For example, to write the ratio of 3 inches to 1 foot, we must first write each
measure in terms of the same unit and then divide each term of the ratio by a
common factor.
In lowest terms, the ratio of 3 inches to 1 foot is 1 : 4.
An equivalent ratio can also be written by multiplying each term of the ratio
by the same non-zero number. For example, 4 : 7 5 4(2) : 7(2) 5 8 : 14.
In general, for x 0:
a : b 5 ax : bx
3 inches
1 foot 5
3 inches
1 foot 3
1 foot
12 inches 5
2-5 RATIO AND PROPORTION
x x 1 2
x x 1 1
x 2 1
x x 2 1
x
Ratio and Proportion 57
DEFINITION
A ratio is the comparison of two numbers by division. The ratio of a to b can
be written as or as a : b when b 0.
a b
EXAMPLE 1
The length of a rectangle is 1 yard and the width is 2 feet. What is the ratio of
length to width of this rectangle?
Solution The ratio must be in terms of the same measure.
Answer The ratio of length to width is 3 : 2.
EXAMPLE 2
The ratio of the length of one of the congruent sides of an isosceles triangle to
the length of the base is 5 : 2. If the perimeter of the triangle is 42.0 centimeters,
what is the length of each side?
Solution Let AB and BC be the lengths of the congruent sides of isosceles ABC and
AC be the length of the base.
AB : AC 5 5 : 2 5 5 x : 2 x
Therefore, AB 5 5 x ,
BC 5 5 x ,
and AC 5 2 x.
AB 1 BC 1 AC 5 Perimeter
5 x 1 5 x 1 2 x 5 42
12 x 5 42
x 5 3.5 cm
Check AB 1 BC 1 AC 5 17.5 1 17.5 1 7.0 5 42.0 cm ✔
Answer The sides measure 17.5, 17.5, and 7.0 centimeters.
5 17.5 cm 5 7.0 cm
AB 5 BC 5 5 (3.5) AC 5 2 (3.5)
1 yd 2 ft
3 ft 1 yd
58 The Rational Numbers
B
A C
5 x 5 x
2 x
