Algebra 2

8-4 Rational Expressions CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Objectives To simplify rational expressions AM-ASFSSE.9A1.2 .AU-sSeStEh.e1s.2tr uUctsuerethoefsatnruecxtupresosfioannto identify
To multiply and divide rational expressions ewxapyrsestosiorenwtoritiedeitn.tify ways to rewrite it.
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evixepwreinssgioonnseboyr vmieowreinogf othneiropr amrtosreasoaf tshinegirlepaerntstitays. a
sAinlsgoleAe-nStSityE..Al.1soa MAFS.912.A-SSE.1.1a

MP 1, MP 2, MP 3, MP 4

The large rectangle and the small X 1

(non-square) rectangle are similar
=ra1ti+os1xl.weniEdgttxhhplaarine
so the equal. Explain X
why x how you can

substitute for x on the right side (only)

of the equation x = 1 + 1x, to get
x = an expression with no x.

Lesson The expression 1 + 1x in the Solve It is equivalent to the rational expression x +x 1.
A rational expression is the quotient of two polynomials. You will find that, at different
Vocabulary
• rational times, it is helpful to think of rational expressions as ratios, as fractions, or as quotients.

expression Essential Understanding  ​You can use much of what you know about
• simplest form multiplying and dividing fractions to multiply and divide rational expressions.

A rational expression is in simplest form when its numerator and denominator are
polynomials that have no common divisors.

In simplest form Not in simplest form

x + 11, x2 + 3x + 2 xx2, 3(x -- 33), x2 - x - 6
x - x +3 x x2 + x - 2

You simplify a rational expression by dividing out the common factors in the numerator
and the denominator. Factoring the numerator and denominator will help you find the
common divisors.

A rational expression and any simplified form must have the same domain in order to
be equivalent.

x2 - x - 6 = (x - 3)(x + 2) and x - 3 , x ≠ - 2, are equivalent.
x2 + x - 2 (x - 1)(x + 2) x - 1

In the example above, you must exclude -2 from the domain of x - 3 because
x - 1
x2 - x - 26 .
-2 is not in the domain of x2 + x - Note that this restriction is not evident from

the simplified expression x - 13.
x -

Lesson 8-4  Rational Expressions 527

Problem 1 Simplifying a Rational Expression

What is x2 + 7x + 10 in simplest form? State any restrictions on the variable.
x2 − 3x − 10

x2 + 7x + 10 = (x + 2)(x + 55)) Factor the numerator and denominator.
x2 - 3x - 10 (x + 2)(x -

Is there more than 5 (x 1 2)(x 1 5) Divide out common factors.
one restriction? (x 1 2)(x 2 5)

Yes, before you divided = x + 5 Simplify.
x - 5
the common factors out,
(x + 2) was one of The simplified form is x + 5 for x ≠ 5 and x ≠ -2. The restriction x ≠ -2 is not evident
the factors of the x - 5
denominator so x ≠ -2. from the simplified form, but is needed to prevent the denominator of the original

expression from being zero.

Got It? 1. What is the rational expression in simplest form? State any restrictions on

the variables.

a. -246xx32yy23 b. xx22 +- 52xx -+ 68 c. 12 - 4x
x2 - 9

You can use what you know about simplifying rational expressions when you multiply
and divide them.

#Problem 2 Multiplying Rational Expressionsx2 − x2
What is the product +x 52 6 x2 + − 25 in simplest form? State any restrictions
x− 4x +
on the variable. 3

How is multiplying # x2 + x -6 x2 - 25
rational expressions x - 52 x2 + 4x + 3
like multiplying #
fractions? = (x + 3)(x - 2) (x + 5)(x - 51)) Factor all polynomials.
x-5 (x + 3)(x +
To multiply rational
(x 1 3)(x 2 2) (x 1 5)(x 2 5)
expressions, you x25 (x 1 3)(x 1 1)
? 5 Divide out common factors.
multiply the numerators
(x - 2)(x + 5)
and multiply the x+1

denominators.

= Simplify.

The product is (x - 2)(x + 5) for x ≠ -3, x ≠ 5, and x ≠ -1. The restrictions x ≠ - 3
and x ≠ 5 are x+1 simplified form, but are needed to prevent the
not evident from the

denominators in the original product from being zero.

# Got It? 2. What is the product x22x--186 xx22 ++ 85xx ++ 4 in simplest form? State any
16
restrictions on the variable.

528 Chapter 8  Rational Functions

To divide rational expressions, you multiply by the reciprocal of the divisor, just as you
do when you divide rational numbers.

Problem 3 Dividing Rational Expressions

What is the quotient x2 2 −x ÷ x2 + 3x − 10 in simplest form? State any
+ 2x + x2 − 1
restrictions on the variable. 1

How do you start? To divide, you multiply by 2 −x ÷ x2 + 3x − 10
Think of division the reciprocal. + 2x + x2 − 1
as multiplying by x2 1
the reciprocal. The expressions may #=
have common factors. 2 −x x2 − 1
So, factor the numerators x2 + 2x + 1 x2 + 3x − 10
and denominators.
#= (x + 1)(x − 1)
Factor -1 from (2 - x) (x 2−x 1) (x + 5)(x − 2)
to get a second (x - 2). + 1)(x +

Divide out common #=
factors. − 1(x − 2) (x + 1)(x − 1)
(x + 1)(x + 1) (x + 5)(x − 2)
Rewrite the remaining
factors. 5 21(x 2 2) ? (x 1 1)(x 2 1)
(x 1 1)(x 1 1) (x 1 5)(x 2 2)
Identify the restrictions
from the denominator of = − 1(x − 1)
the simplified expression (x + 1)(x + 5)
and from any other
denominator used. x ≠ −1, x ≠ −5, x ≠ 1, and x ≠ 2

Got It? 3. a. What is the quotient xx22 ++ x5x-+124 , 2xx22 - 1 in simplest form? State any
- 6x
restrictions on the variable.

b. Reasoning  Without doing the calculation, what is greatest number of

restrictions the quotient x2 + 8x + 7 , x2 + 2x - 8 could have? Explain.
x2 - x - 12 x2 + 13x + 24

Lesson 8-4  Rational Expressions 529

Problem 4 Using Rational Expressions to Solve a Problem STEM

Construction  Your community is building a park. It wants to fence in a play space for
toddlers. It wants the maximum area for a given amount of fencing. Which shape,
a square or a circle, provides a more efficient use of fencing?

One measure of efficiency is the ratio of area fenced to fencing used, or area to perimeter.
Which of the two shapes has the greater ratio?

Square Circle
Area = s2 Define area and perimeter. Area = pr2

Perimeter = 4s Perimeter = 2pr

s = P Express s and r in terms of a r = P
4 common variable, P. 2p

Area = sP2 Write the ratios. Area = pr 2
Perimeter Perimeter P

( )P 2 ( )Substitute for s and r. P 2
2p
= 4 = p
P
How can you P
ce4Ppovmawlpuitaahrtoein u1gPt6Pa?nd
Since the numerators = 1P6 Simplify. = P
are the same, the 4p
fraction with the smaller
denominator is the larger Since P 7 1P6, a circle provides a more efficient use of fencing.
fraction. 4p

( )Check Assume P = 40 ft. The area of the circle is p 40 2 ≈ 127 ft2.
2p
( )The area of the square is
40 2 = 100 ft2. The area of the circle is greater.
4

Got It? 4. Which shape of play space provides for a more efficient use of fencing, a
square or an equilateral triangle? (Hint: The area of an equilateral triangle in

( )terms of one side is 12(s) 123s . The perimeter of an equilateral triangle is 3s.)

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

Simplify each rational expression. State any restrictions 5. Vocabulary  ​Is the equation y = x + 1 in simplest
on the variables. form? Explain how you can tell. x2 + 1

1. 48zz - 1224 2. 3x - 3 6. Error Analysis  ​A student claims that x = 2 is the
+ x2 - x x = 2
only solution of the equation x - 2 x - 2. Is the

Multiply or divide. State any restrictions on the student correct? Explain.

variables. 7. Reasoning  ​The width of the rectangle
a + 10
# 3. xx22 + 3x - 10 3x + 18 is 3a + 24 . Write an expression for 2a ϩ 20 w
+ 4x - 12 x+3 3a ϩ 15
the length of the rectangle in
ᐍ
4. xx22 - 7x + 10 , 4 - x2 simplest form.
- 8x + 15 + 3x -
x2 18

530 Chapter 8  Rational Functions

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Simplify each rational expression. State any restrictions on the variables. See Problem 1.

8. - 155xx3yy3 9. 2x 2x
10. 6c23+c 9c 4x2 -
12. xx22 +- 28xx +- 1246
11. 49 - z2
z + 7

13. 12 - x - x2
x2 - 8x + 15

Multiply. State any restrictions on the variables. #15. 5y3 See Problem 2.
2x4 4x3 See Problem 3.
# 14. 45xy2 172yx4 10y-2
# 16. 180yy--45 53yy--195 #17. 23xx
# 18. xx22 -- 14 xx2 ++ 21x + 12 6 - 2x
- 9 3x + 8
Divide. State any restrictions on the variables. #19. x2
20. 47yx3 , 281yx3 - 5x + 6 x2 + 3x + 2
22. 6xy -+ x6y , 5x 1-8 5y x2 - 4 x2 - 2x - 3
24. x2 + x22x + 1 , x23-x 1
21. 3x3 , 6y-3
5y2 5x-5

23. 32yy - 12 , 6y - 24
+4 8 + 4y

25. y2 - 5y + 6 , y2 + 3y - 10
y3 4y2

STEM 26. Industrial Design  A​ storage tank will have a circular base of radius r and a See Problem 4.

height of r. The tank can be either cylindrical or hemispherical (half a sphere).

a. Write and simplify an expression for the ratio of the volume of the hemispherical
43pr3
tank to its surface area (including the base). For a sphere, V = and
SA = 4pr2.
b. Write and simplify an expression for the ratio of the volume of the cylindrical

tank to its surface area (including the bases).

c. Compare the ratios of volume to surface area for the two tanks.

d. Compare the volumes of the two tanks.

e. Describe how you used these ratios to compare the volumes of the two tanks.

Which measurement of the tanks determines the volumes?

B Apply Simplify each rational expression. State any restrictions on the variables.

27. xx22 -- 75xx -- 3240 28. 2y2 + 8y - 24 29. xy3 - 9xy 144x
2y2 - 8y + 8 12xy2 + 12xy -

30. Open-Ended  Write three rational expressions that simplify to x x 1.
+

Lesson 8-4  Rational Expressions 531

31. Think About a Plan  ​A cereal company wants to use the most efficient packaging
for their new product. They are considering a cylindrical-shaped box and a cube-
shaped box. Compare the ratios of the volume to the surface area of the containers
to determine which packaging will be more efficient.

• How can you measure the cereal box’s efficiency?
• What formulas will you need to use to solve this problem?

Multiply or divide. State any restrictions on the variables.
#33.
32. 6xx43+-56xx32 , 32xx22-+125xx + 12 x2 2x2 - 6x 81 9x + 81
- 40 + 18x + x2 - 9
#35.
34. 2xx22 -- x5x-+22 , 2xx22-+ x- 12 2x2 + 5x + 2 2x2 + x - 1
5x -3 4x2 - 1 x2 + x - 2

36. a. Reasoning W​ rite a simplified expression for the area of the rectangle at 4a ϩ 4
aϩ3
the right.
b. Which parts of the expression do you analyze to determine the restrictions 3a ϩ 9
2a Ϫ 6
on a? Explain.
c. State all restrictions on a.

STEM 37. Manufacturing  A toy company is considering a cube or sphere-shaped
container for packaging a new product. The height of the cube would
equal the diameter of the sphere. Compare the volume-to-surface area ratios of the
containers. Which packaging will be more efficient? For a sphere, SA = 4pr 2.

Decide whether the given statement is always, sometimes, or never true.

38. Rational expressions contain exponents.

39. Rational expressions contain logarithms.

40. Rational expressions are undefined for values of the variables that make the
denominator 0.

41. Restrictions on variables change when a rational expression is simplified.

Simplify. State any restrictions on the variables.

42. x(x - 1)(-x22(-x2x+)23x - 4) 43. (x - 2x + 6 2x - 3) 44. 54x3y-1
1)-1(x2 + 3x-2y

C Challenge 45. a. Reasoning  S​ implify (2xn)2 - 1 , where x is an integer and n is a positive integer.
2xn - 1
(Hint: Factor the numerator.)

b. Use the result from part (a). Which part(s) of the expression can you use to show

that the value of the expression is always odd? Explain.

a
b a c
Use the fact that c = b ÷ d to simplify each rational expression. State any

d
restrictions on the variables.

8x2y 3a3b3 9m + 6n x2 - 1
a-b m2n2 x2 - 9
46. x6x+y21 47. 4ab 48. 12m + 8n 49. x2 + 3x - 4

x+1 b-a 5m2 x2 + 8x + 15

532 Chapter 8  Rational Functions

Standardized Test Prep

SAT/ACT 50. Which function is graphed at the right? 8y x
4
y = (x + 4)(x - 1)(x + 2) Ϫ2 O
y = (x - 4)(x - 1)(x + 2) Ϫ8
y = (x - 4)(x + 1)(x - 2)
y = (x + 4)(x + 1)(x - 2)

51. Which function generates the table of values at the right? xy

y = log1x 1 Ϫ1
2 2

y = - log2x 10
y = log2x
21
( ) y = 1x
2 42

# x 2x - 6
2x - 4x +

52. Which expression equals x2 - - 3 x2 3?

(x - 2x - 1 + 1)
1)(x + 3)(x

(x - 2x + 1 - 3)
1)(x + 1)(x

(x - 1)(x 2x 1)(x - 3)
+

(x + 3)(x 2x 1)(x + 1)
-

RS heosprtonse 53. What is the solution of the equation 3-x = 1 ?
243

Mixed Review

Find the vertical asymptotes and holes for the graph of each rational function. See Lesson 8-3.

54. y = xx -- 33 55. y = (3x x-1 + 1) 56. y = (x - 4)(x + 5)
+ 2)(x (x + 3)(x - 4)

Evaluate each logarithm. See Lesson 7-3.
60. log16 8
57. log4 64 58. log2 312 59. log5 5 15

Solve. Check for extraneous solutions. See Lesson 6-5.

61. 1x - 3 = 4 62. 1x + 1 - 5 = 8 63. 15x - 3 = 12x + 3

G et Ready!  To prepare for Lesson 8-5, do Exercises 64–67.

Add or Subtract. See p. 973.

64. 159 + 378 65. 2 + 235 66. 274 - 356 67. 1112 - 7
15 45

Lesson 8-4  Rational Expressions 533

8-5 Adding and Subtracting CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Rational Expressions
MA-AAFPSR.9.D1.27. A.-.A. PARd.d4,.7su  b. t.r.acAtd, md,uslutibptlrya, catn, dmduilvtipdley,
raantdiodniavlideexpraretisosnioanl se.xpressions.

MP 1, MP 3, MP 4

Objective To add and subtract rational expressions

You can use a table At 3 p.m., four runners all leave the
to organize the starting line, running laps around the
information. indoor track. If the runners maintain
their pace, at what time will Sue,
MATHEMATICAL Drew, and Stu finish a lap together?
At what time will all four runners finish
PRACTICES a lap together? Explain your reasoning.

Lesson You use common multiples of polynomials to add and subtract rational expressions,
Vocabulary just as you use common multiples of numbers to add and subtract fractions.
• complex fraction
Essential Understanding  To operate with rational expressions, you can use
much of what you know about operating with fractions.

To add or subtract rational expressions, you first find a common denominator—
preferably the least common multiple (LCM) of the denominators.

To find the LCM of several expressions, factor the expressions (numbers or
polynomials) completely. The LCM is the product of the prime factors, each raised
to the greatest power that occurs in any of the expressions.

Problem 1 Finding the Least Common Multiple

How do you What is the LCM of 12x2y(x2 + 2x + 1) and 18xy3(x2 + 5x + 4)?
determine the
exponent of each Step 1 Find the prime factors of each expression.
factor for the LCM?
#12x2y(x2 + 2x + 1) = 22 3x2y(x + 1)2
Use the exponent from #18xy3(x2 + 5x + 4) = 2 32xy3(x + 1)(x + 4)

the expression that Step 2 Write the product of the prime factors, each raised to the greatest power
that occurs in either expression.
has that factor to the
#22 32x2y3(x + 1)2(x + 4)
greatest power. #The LCM is 22 32x2y3(x + 1)2(x + 4), or 36x2y3(x + 1)2(x + 4).

534 Chapter 8  Rational Functions

Got It? 1. What is the LCM of the expressions?
a. 2x + 4 and x2 - x - 6
b. x2 + 3x - 4, x2 + 2x - 8, and x2 - 4x + 4

the LCM of the denominators of two rational expressions is also the Least Common
Denominator (LCD) of the rational expressions. You can use the LCD to add or subtract
the expressions.

Recall how you used the LCD to add fractions.

# #1 5 22 5 9
5 22 40 40
8
+ 1 = 1 +215 = 213 a b + 1 5 a b = + 4 =
10 23 40
2

Problem 2 Adding Rational Expressions

What is the sum of vthaeritawboler.axti−ox n1al+exxp2r2−exs3s−ixo1+ns2in simplest form? State any
restrictions on the

How does the LCD x 2x - 1 x 2x - 1
- x2 - 3x + - - 1)(x -
help you simplify this x 1 + 2 = x 1 + (x 2) Factor the denominators.
Rewrite each expression with the LCD.
sum? #=
Add the numerators. combine like terms.
The LCD is (x - 1)(x - 2). x x - 2 + 2x - 1 2)
Multiply the first - x - 2 - 1)(x -
etoxpgreetssaiocnombymxxon-- 2 x 1 (x
2
x2 - 2x 2x - 1
denominator. = (x - 1)(x - 2) + (x - 1)(x - 2)

= x2 - 2x + 2x - 1
(x - 1)(x - 2)

= (x x2 - 1 2)
- 1)(x -

= (x - 11)) ((xx + 1) cf aocmtomrotnhefancutomres.rator and divide out the
(x - - 2)

= x + 12, x ≠ 1
x -
x + 1
The sum of the expressions is x - 2 for x ≠ 1 and x ≠ 2.

Got It? 2. What is the sum of the two rational expressions in simplest form? State any
restrictions on the variable.

a. xx + 1 + -2 x
- 1 x2 -
x 1
b. x2 - 4 + x + 2

c. Reasoning  Is it possible to add the rational expressions in Problem 2

by finding a common denominator, but not the least common

denominator? Explain.

Lesson 8-5  Adding and Subtracting Rational Expressions 535

Problem 3 Subtracting Rational Expressions

What is the difference of the two rational expressions in simplest form? State any
x + 2 x+2
restrictions on the variable. x2 − 2x − 2x − 4

How is this problem x + 2 - x+2 = x+2 - x +- 22) Factor the denominators.
similar to Problem 2? x2 - 2x 2x - 4 x(x - 2) 2(x
The method is the same
except you subtract the The LCD is 2x(x - 2).
rational expressions # #=
instead of adding them. x+2 2 - x+2 x wReitwhrtitheeeLaCcDh. expression
x(x - 2) 2 2(x - 2) x

= 2(x + 2) - x(x + 2)
2x(x - 2) 2x(x - 2)

= 2x + 4 - x2 + 2x Simplify the numerators.
2x(x - 2) 2x(x - 2)

= 2x +4- (x2 + 2x) Subtract the numerators.
2x(x - 2)

= - x2 + 24) combine like terms.
2x(x -

= - (x2 - 4) Factor -1 from the numerator.
2x(x - 2)

= - (x - 2)(x + 2) Factor x2 - 4 and divide out the common factors.
2x(x - 2)

= - (x + 2)
2x

The difference is - (x + 2) for x ≠ 2 and x ≠ 0.
2x

Got It? 3. What is the difference of the two rational expressions in simplest form?

State any restrictions on the variable.

a. xx + 3 - x2 6x - 7 2 b. x - 1 - x2 x +3 5
- 2 - 3x + x + 5 + 6x +

A complex fraction is a rational expression that has at least one fraction in its
numerator or denominator or both. Here are some examples.

1 + 1     x + 3       x2 x +3 1 + x2 - x 2
x 1 y x 2 4 - 2x + - x2 3x +
xy 2 2
- x -4
x2 - 4x + 4

536 Chapter 8  Rational Functions

Sometimes you can simplify a complex fraction by multiplying the numerator and the
denominator by the LCD of all the rational expressions. You can also simplify them by
combining the fractions in the numerator and those in the denominator. Then multiply
the new numerator by the reciprocal of the new denominator.

Problem 4 Simplifying a Complex Fraction

What is a simpler form of the complex fraction?

1 + x
x y

1x, 1 + 1
y
What is the LCD of
yx, and 1y? Method 1 Multiply both the numerator and the denominator by the LCD of all the

The LCD of the rational rational expressions and simplify the result.
1
expressions is xy. + x 1 + x xy
( ) #xy x y
= Multiply the numerator and the denominator by xy.
1+1 1 + 1 xy Use the Distributive Property.
y
( ) #y
# #=
1 xy + x xy
x y
1# #y
xy + 1 xy

= y + x2 Simplify.
x + xy

Method 2 Combine the expressions in the numerator and those in the denominator.

Then multiply the new numerator by the reciprocal of the new denominator.
1
+ x = 1 y + x x
# #xy x y y y x cW ormitme eoqnudiveanloemntineaxtporress.sions with
y
1+1 1 +1
y
#y

= y + x2 Multiply.
xy + xy
1 y
y y

y + x2
xy
= 1+y Add.

How do you divide y
a fraction by a
fraction? = y + x2 , 1 + y Divide the numerator fraction by the
xy y denominator fraction.
Multiply the numerator
#= y + x2 y y Multiply by the reciprocal.
by the reciprocal of the xy + Divide out the common factor, y.
1
denominator.
y + x2
= x + xy

Got It? 4. What is a simpler form of the complex fraction?
x - 2 2
a. 1x +x 1y b. x x 1 + x + 1
3 - x 1 1
- +

Lesson 8-5  Adding and Subtracting Rational Expressions 537

Problem 5 Using Rational Expressions to Solve a Problem

Fuel Economy  A woman drives an SUV that gets 10 mi/gal (mpg). Her husband
drives a hybrid that gets 60 mpg. Every week, they travel the same number of miles.
They want to improve their combined mpg. They have two options on how they can
improve it.

Option 1:  They can tune the SUV and increase its mileage by 1 mpg and keep the
hybrid as it is.

Option 2:  They can buy a new hybrid that gets 80 mpg and keep the SUV as it is.

Which option will give them a better combined mpg?

The combined gas combined mpg = SUV miles + Hybrid miles
mileage is total miles SUV gallons + Hybrid gallons
divided by total gallons.
Let x = number of miles each drives in a week.
Define a variable and
describe each option. Option 1 Option 2

The gallons used by each Tuned SUV gets 11 mpg. SUV gets 10 mpg.
vehicle are mmiplegs. Write Hybrid gets 60 mpg. New hybrid gets
the variable expressions 80 mpg.
for each option’s
combined mpg. x+x x+x

Find the LCD of the x + x x + x
fractions in each 11 60 10 80
expression. Multiply
the numerator and ° # ( )2x¢ 660 # ( )2x¢ 80
denominator by the LCD. x + x 660 ° x + x 80
11 60 10 80
Distribute and simplify.
(2x)(660) (2x)(80)
Round the ratios and
compare them. ( ) ( )x x ( ) ( )x x
11 60 10 80
(660) + (660) (80) + (80)

= 1320x = 160x
60x + 11x 8x + x

= 1320x = 160x
71x 9x

? 18.6 mpg ? 17.8 mpg

Option 1 gives the better combined mpg.

Got It? 5. Suppose Option 3 is to buy a new hybrid that will get double the mileage
of the present hybrid. The SUV mileage stays the same. Which of the three
options will give the best combined mpg?

538 Chapter 8  Rational Functions

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

Simplify each sum or difference. State any restrictions 5. Error Analysis  ​Describe and correct the error made
on the variables.
in simplifying the complex fraction.

1. 3aa+-151 + a - 21 1 x+ 1
3a - 5 x x
1 + 3 3( x + 1)
1 6 = x = x2
2. x2 - + + 3
4 x 2 x

3. 3mm+ 6 - 4m 6. Open-Ended  W​ rite two rational expressions that
m+2
+
4. b2 b -4 - b + 2 simplify to x - 51.
+ 2b - b2 - 16 x
8

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find the least common multiple of each pair of polynomials. See Problem 1.

7. 9(x + 2)(2x - 1) and 3(x + 2) 8. x2 - 1 and x2 + 2x + 1

9. 5y2 - 80 and y + 4 10. x2 - 32x - 10 and 2x + 10

Simplify each sum or difference. State any restrictions on the variables. See Problems 2 and 3.

11. 21x + 21x 12. d - 3 + d- 11 13. -x2 - 1
2d + 1 2d + x

14. 2y- -5y1 - 2yy+-31 15. 5y + 2 + 2x - 4 16. 5x + 2
17. 2y y+ 4 - y +3 2 xy2 4xy x2 - +
20. x2 -5xx - 6 + x2 + 44x + 4 9 x 4

18. 3x x 9 - x2 8 3x 19. - 3x + 2x 4 6
+ + x2 -9 -

21. x2 2x - 2 - x2 4x + 2
-x - 3x

Simplify each complex fraction. See Problem 4.

1 1 - 1 2 1
22. 2x 2 - 4 x + y
23. 3 24. 3 25. 3
y 5 3
b
2 3
26. 1 +1 xy 3 x + y
27. + 28. 5 29. x - 4
2 y 1 - x 2
x x+y - 4

30. Your car gets 25 mi/gal around town and 30 mi/gal on the highway. See Problem 5.
a. If 50% of the miles you drive are on the highway and 50% are around town, what

is your overall average miles per gallon?
b. If 60% of the miles you drive are on the highway and 40% are around town, what

is your overall average miles per gallon?

Lesson 8-5  Adding and Subtracting Rational Expressions 539

B Apply Add or subtract. Simplify where possible. State any restrictions on the variables.

31. 43x - x22 32. x 3 1 + x x 1 33. x2 4 9 + x 7 3
+ - - +

34. x2 -5xx - 6 - x2 + 44x + 4 35. 3x + x2 + 5x 36. y2 5y - 2y 4 14 + 9
x2 - 2 - 7y - y

37. Think About a Plan  For the image of the overhead projector to be in focus, the
distance di from the projector lens to the image, the projector lens’s focal length f,
tathrnaidnn-stlpheaenrdseinesqctauynapctlieaocdneod1ffr=4oimdn1i.+tfhrode1mot.rWathnhesapptarirosejtnehccetyoftorocleathnl elsepinsrgiontjhefcootcfoutrhsleeonpnsrtomhjeeucsstcotrsrealeetninsfslyoicfthathteeed
8 ft from the projector lens?

• Can you write the equation for the unknown variable?
• What units would you use for the focal length of the lens?

STEM 38. Optics  To read a small font, you use a magnifying lens with the focal length 3 in. How
far from the magnifying lens should you place the page if you want to hold the lens
at 1 foot from your eyes? Use the thin-lens equation from Exercise 37.

39. Reasoning  Does the Closure Property of rational numbers extend to rational
expressions? Explain and describe any restrictions on rational expressions.

40. Writing  E​ xplain how factoring is used when adding or subtracting rational
expressions. Include an example in your explanation.

Simplify each complex fraction.

41. -2xx5++3y7y 42. 1+ 2 43. 1 - 1 44. x 2 4 + 2
2+ x xy - y2 1 + x 3 4
3 1 1 + +
2x x2y xy2

STEM 45. Harmony  ​The harmonic mean of two numbers a and b equals 1 2 1 . As you vary
a +
b

the length of a violin or guitar string, its pitch changes. If a full-length string is 1 unit

long, then many lengths that are simple fractions produce pitches that harmonize,

or sound pleasing together. The harmonic mean relates two lengths that produce

harmonious sounds. Find the harmonic mean for each pair of string lengths.

a. 1 and 12 b. 3 and 1 c. 3 and 53 d. 1 and 1
4 2 4 2 4

C Challenge 46. Show that the sum of the reciprocals of three different positive integers is greater
than 6 times the reciprocal of their product.

STEM 47. Electricity  The resistance of a parallel circuit with 3 bulbs is 1 + 1 + 1.
R1 1
R2 R3

a. Find the resistance of a parallel circuit with 3 bulbs that have resistances 5 ohms,

4 ohms, 2.5 ohms.

b. The resistance of a parallel circuit with 3 bulbs is 1.5 ohms. Find the resistances

of the bulbs, if two of them have equal resistances, while the resistance of the 3rd

is 3 ohms less.

540 Chapter 8  Rational Functions

Standardized Test Prep

SAT/ACT 48. Which expression equals 5x 9 - x2 4x + 6?
x2 - + 5x
(x 7x 2) (x x2 + 22x
- 3)(x + 3)(x + - 3)(x + 3)(x + 2)

(x - x2 - 2x + 2) (x - 9x2 - 2x + 2)
3)(x + 3)(x 3)(x + 3)(x

49. Which of the relationships is represented by the graph at the right? y x
y = log4(x - 1) + 5 4
y = log4(x - 1) - 2 O2
y = log4(x + 2) - 2 Ϫ2
y = log4(x - 1) - 1
Ϫ4

50. What is a simpler form of  2 - 5 ?
x - 3
6
x

62 - 5x 62 + 5x 2x - 5 26 + 3x
- 3x - 3x 6x + 3 - 5x

51. What word makes the statement “The domain and range of a(n) ________________
function is the set of all real numbers” sometimes true?

polynomial exponential

logarithmic quadratic

S hort 52. What is the least common denominator for the rational expressions x2 - 1 - 6 and
Response + 36? Show your work. 5x
1
x2 - 12x

PERFORMANCE TASK MATHEMATICAL

Apply What You’ve Learned PRACTICES
MP 2

Look back at the information on page 497 about Steve and Sally’s canoe trip.
In the Apply What You’ve Learned in Lesson 8-2, you thought about how the
current of the river affects the speed Steve and Sally travel while paddling
upstream.

Let x represent the speed, in miles per hour, that Steve and Sally paddle in still
water. Select all of the following expressions that represent the total amount of
time Steve and Sally spent paddling. Explain your reasoning.

A. x 1-2 3 B. 12 + 12
x-3 x

C. x 1-2 3 + x 1+2 3 D. (x - 12 + 3)
3)(x

E. x224-x 9 F. 24x
x2 - 6x - 9

Lesson 8-5  Adding and Subtracting Rational Expressions 541

8-6 Solving Rational MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Equations
MA-AAFPSR.9.D1.27. A.-.A. PARd.d4,.7su  b. t.r.acAtd, md,uslutibptlrya, catn, dmduilvtipdley, and
rdaivtiodnearlaetixopnraelsesxiopnres.ssAiolsnos. A-lAsoPRM.DA.F6S,.A91-C2.EAD-.AAP.1R,.4.6,
AM-ARFESI..D91.121.A-CED.1.1, MAFS.912.A-REI.4.11

MP 1, MP 2, MP 3, MP 4, MP 5

Objectives To solve rational equations
To use rational equations to solve problems

You normally walk the sidewalk from home 0.5 mi
to school in 25 minutes. If you walk the
shortcut at your normal constant rate, how School
much time would you save? Show how you
Use units to found your answer.
make sure you set
the problem up 1.2 mi
correctly.

MATHEMATICAL

PRACTICES

Home

Lesson Sometimes you can solve a problem using a proportion—an equation involving two
rational expressions set equal to each other.
Vocabulary
• rational equation Essential Understanding  To solve an equation containing rational expressions,
first multiply each side by the least common denominator of the rational expressions.
Doing this, however, can introduce extraneous solutions.

A rational equation contains at least one rational expression. You can simplify solving
a rational equation if you first clear the equation of denominators. You can do this by
multiplying by the LCD of the rational expressions in the equation.

Rational Equation Not a Rational Equation

x x 1 + x x 1 = 2 1 x + 1 = 2
+ - x2 - 2 3

Any time you multiply each side of an equation by an algebraic expression, it is possible
to introduce an extraneous solution. Recall that an extraneous solution is a solution of
the derived equation, but not a solution of the original equation. You must check all
solutions in the original equation to confirm that they are indeed solutions.

542 Chapter 8  Rational Functions

Problem 1 Solving a Rational Equation

What are the solutions of the rational equation?
How can you use
t echnology to solve A x x 3 + x x 3 = x2 2 9
this equation? − + −
You can use a computer
algebra system to solve factor the  x x 3 + x x 3 = (x − 2 + 3)
the equation. Calculators denominators to − + 3)(x
have limitations, and you find the LCD.
will still need to check for (x − 3)(x + 3) J x x 3 + x x 3R = (x − 3)(x + 3) (x − 2 + 3)
extraneous solutions. Multiply each side − + 3)(x
by the LCD to clear
denominators. x2 + 3x + x2 − 3x = 2
2x2 = 2
Now simplify
and solve. x2 = 1, so x = t 1

Check whether 1 1 3 + 1 1 3 ≟ 2     −1 3 + −1 ≟2 − 9
x = 1 or − + (1)2 − −1 − −1 + 3 ( − 1)2
x = -1 is 9
extraneous.
Use the original − 1 + 1 = − 1 ✔ 1 + − 1 = − 1 ✔
equation. 2 4 4 4 2 4

Write the The solutions are x = 1 and x = −1.
solutions.

B x2 x −1 2 + x 2x 2 = x − 1
+ 3x + + x + 1

Use a computer algebra system (CAS) to solve this rational equation.

Step 1 On the Home screen, Step 2 Choose Menu, Step 3 Enter the equation,

choose New Document. Algebra, Solve. followed by a comma

Then select and x. Then press

Add Calculator. enter to solve.



The calculator shows a warning because there may be extraneous solutions. The
original equation restricts x so that x ≠ -2 and x ≠ -1. There is no solution.

Got It? 1. What are the solutions of the rational equation?

a. xx -+ 21 = x2 +x +2x2- 3 b. x x 1 + x 3 4 = x + 3
+ + x + 4

Lesson 8-6  Solving Rational Equations 543

Problem 2 Using Rational Equations

Flight  A flight across the U.S. takes 1850 mi Chicago
longer east to west than it does west San Francisco
to east. Assume that winds are constant
in the eastward direction. When flying
westward, the headwind decreases the
airplane’s speed. When flying eastward,
the tailwind increases its speed. The
time for a round trip shown at the right
is 734h. If the airplane cruises at 480 mi/h,
what is the speed of the wind?

Let x = the wind speed.

Rate * Time = Distance, so Time = Distance
Rate

Going west to east Distance Rate Time
Going east to west 1850 480 ϩ x
1850 480 Ϫ x 1850
480 ϩ x

1850
480 Ϫ x

Total time = Time west to east + Time east to west Multiply both sides
by the LCD,
7.75 = 1850 x + 1850 (480 + x)(480 − x).
480 + 480 - x

(480 + x)(480 - x) 7.75 = (480 + x)(480 - x) 1850 x + (480 + x)(480 - x) 1850 x
480 + 480 -

7.75(480 + x)(480 - x) = 1850(480 - x) + 1850(480 + x)

1,785,600 - 7.75x2 = 888,000 - 1850x + 888,000 + 1850x

- 7.75x2 = - 9600

x2 = - 9600
- 7.75

If you substitute x ≈ {35
35 for x will the
equation check Wind speed is positive, so x ≈ 35. The west-to-east wind speed is about 35 mi/h.
exactly?
Check 7.75 = 1850 x + 1850
No; since 35 is an 480 + 480 - x

approximation it is 7.75 ≟ 1850 + 1850
480 + 35 480 - 35
likely that the values
7.75 ≈ 3.6 + 4.2 ✔
will be nearly equal, but

probably not equal.

Got It? 2. a. You ride your bike to a store, 4 mi away, to pick up things for dinner.
When there is no wind, you ride at 10 mi/h. Today your trip to the store

and back took 1 hour. What was the speed of the wind today?
b. Reasoning  Explain why there is no difference between the travel time to

and from the store when there is no wind.

544 Chapter 8  Rational Functions

You can also use a graphing calculator to solve a rational equation.

Problem 3 Using a Graphing Calculator to Solve a Rational Equation

What are the solutions of the rational equation? Use a graphing calculator to solve.

x 2 2 + x x 2 = 1
+ −

How do the graphs of Enter one side of the There appears to be Y1 ϭ Y2
the two sides of the equation as Y1. Enter only one intersection when x ϭ 0.
equation help you the other side as Y2. point, at x ϭ 0.
solve the equation?
The x-values of the Plot1 Plot2 Plot3 X Y1 Y2
points of intersection \Y1 = (2/(X+2))+(X/(X–2))
\Y2 = 1 –1 2.3333 1
are the solutions to the \Y3 = 0 1 1
\Y4 = 1 –.3333 1
equation. \Y5 = 2 ERROR 1
\Y6 = 3 3.4 1
\Y7 = 4 2.3333 1
5 1.9524 1

X=0

The solution is x = 0.

Check x 2 2 + x x 2 = 1
+ -

0 2 2 + 0 0 2 ≟ 1
+ -

1 + 0 = 1 ✔

Got It? 3. What are the solutions of the rational equation x+2 = 5?
1 - 2x
Use a graphing calculator to solve.

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Solve each equation. Check each solution. 5. Error Analysis  D​ escribe and
PRACTICES
correct the error made in
1. x 4 = x - 1 solving the equation. 5 + 9 = 28
- x - 2 x 7 x
6. Open-Ended  ​Write a rational
2 equation using expressions 14 = 28
that have x2 - 9 as their LCD. x+7 x
2. 2a + 1 a a - 1
6 + 2 = 3 7. Reasoning  ​Describe two 14x = 28( x + 7)
methods you can use to 14 x = 28x + 196
3. n2 + n + 2 = -2 n check whether a solution −196 = 14x
n + 1 n2 + is extraneous. −14 = x

4. Flight  If the speed of an airplane is 350 mi/h with a
tail wind of 40 mi/h, what is the speed of the plane in
still air?

Lesson 8-6  Solving Rational Equations 545

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Solve each equation. Check each solution. See Problem 1.

8. 41 - x = 8x 9. y + y = 7 10. 2x - 1 = 2x + 5
5 2 3 2 6

11. 3x1-2 2 - 16 = 16 12. 1 + x = x + 4 13. 11 - 1 = -4
14. 23x - 35x = 2 x 2 2x 3x 3 x2

15. 5x - 3 = 41 16. 2 + 1 = 5
4 x y 2 2y

17. x + 6x = - 5 18. 1 - 3 = x7 19. 5 - 2 = 1 + 5
4x 4 2x 3 x 6

20. Transportation  The speed s of an airplane is given by s = dt , where d represents See Problem 2.
the distance and t is the time.

a. A plane flies 700 miles from New York to Chicago at a speed of 360 mi/h. Find

the time for the trip.
b. On the return trip from Chicago to New York, a tail wind helps the plane move faster.

The total flying time for the round trip is 3.5 h. Find the speed x of the tail wind.

Graphing Calculator  solve each equation. Check each solution. See Problem 3.

21. 3x = 5 22. 1 = -2 23. x 2 1 = 4
3x -

24. x +4 3 = 5 25. 5x - 2 = -3 26. 3x - 1 = 7
27. 2x = 2x x- 4 x+ 2

28. x 2 3 = x - 3 29. x 2 1 + x 3 1 = 4
+ 2 - +

B Apply Solve each equation for the given variable.

30. m = V2E2; E 31. c - 1 = 0; E 32. m = 1 ; F
E mc F a

33. 1c - a2 -c b2 = 0; c 34. / = g ; T 35. q = 2V ; B
T2 4p 2 m B2r 2

36. Think About a Plan  Y​ ou and a classmate have volunteered to contact every
member of your class by phone to inform them of an upcoming event. You can
complete the calls in six days if you work alone. Your classmate can complete
them in four days. How long will it take to complete the calls working together?

• If N is the total number of calls, what expression represents the number of calls
that you can make per day? What expression represents the number of calls your
friend can make per day?

• What is the expression for the number of days needed to make N calls if you are
working together?

37. Storage  One pump can fill a tank with oil in 4 hours. A second pump can fill the
same tank in 3 hours. If both pumps are used at the same time, how long will they
take to fill the tank?

546 Chapter 8  Rational Functions

38. Teamwork  You can stuff envelopes twice as fast as your friend. Together, you can
stuff 6750 envelopes in 4.5 hours. How long would it take each of you working alone
to complete the job?

39. Grades  On the first four tests of the term your average is 84%. You think you can
score 96% on each of the remaining tests. How many consecutive test scores of 96%
would you need to bring your average up to 90% for the term?

40. Error Analysis  ​Describe and correct the x −x 2 2 = x+1
error made in solving the equation. − x+2

41. Fuel Economy  ​Suppose you drive an x − 2(x + 2) = (x + 1)(x − 2)
average of 15,000 miles per year, and your
car gets 24 miles per gallon. Suppose x − 2x − 4 = x2 −x − 2
gasoline costs $3.60 a gallon. 0 = x2 + 2
−2 = x2

a. How much money do you spend each

year on gasoline? There is no square root of a

b. You plan to trade in your car for one negative number, so the

that gets x more miles per gallon. Write equation has no solution.

an expression to represent the new

yearly cost of gasoline.

c. Write an expression to represent your total savings on gasoline per year.

d. Suppose you can save $600 a year with the new car. How many miles per gallon

does the new car get?

STEM 42. Woodworking  ​A tapered cylinder is made by decreasing the

radius of a rod continuously as you move from one end to the r

other. The rate at which it tapers is the taper per foot. You can R

calculate the taper per foot using the formula T = 24(R - r).
L
The lengths R, r, and L are measured in inches.
L
a. Solve this equation for L.

b. What is L for T = 0.75, 0.85, and 0.95, if R = 4 in.; r = 3 in.?

Solve each equation. Check each solution. 44. x 2 2 - 1 = -4 2)
43. 1x5 + 9xx+-27 = 9 + x x(x +

45. b +1 1 + b -1 1 = b2 2 1 46. c - c + c = 26
- 3 5

47. x -1 5 = x 2 -x 25 48. k k 1 + k k 2 = 2
+ -

49. x 2 - 75x + 12 - 3 -2 x = x 5 4 50. 2y1+0 8 - 7y + 8 = -8 8
- y 2 - 16 2y -

51. x 2 -7x8+x +3 15 + x 3-x 5 = 3 1 x 52. x 2 3 - 4 3 x = 2x -2
- + - x2 - x - 12

53. Writing  ​Write and solve a problem that can be modeled by a rational equation.

Lesson 8-6  Solving Rational Equations 547

C Challenge 54. Sports  An automatic pitching machine can pitch all its baseballs in 1 1 hours.
One attendant can retrieve all the baseballs pitched by one machine 4 1
in 3 2 hours.

At least how many attendants working at the same rate should be hired so that the

baseballs from 10 machines are all retrieved in less than 8 hours?

55. Open-Ended  ​Write a rational equation that has the following.

a. one solution b. two solutions c. no real solution

STEM 56. Industry  ​The average hourly wage H (x) of workers in an industry is modeled
bsiyntchee 1f9u7n0c.tIinonwHha(txy)e=ar0d.0o6e21sx6.t2+h4e3x9m.4o2d, welhperreedxicrtetphraetswenatgsetshwe inllubmeb$e2r5o/hf ?years

Standardized Test Prep

SAT/ACT 57. What is the solution of x + 1 = -2?
x

1, -1 0 only - 1 only -1 only
2 6 12

58. Which of the following is equivalent to 6 124 ? 200%
213
2 12 3 12 5 12

59. An investment of $750 will be worth $1500 after 12 years of continuous
compounding at a fixed interest rate. What is that interest rate?

2.00% 5.78% 6.93%

E xtended 60. A librarian orders 48 fiction and nonfiction books for the school library. A fiction
Response book costs $15 and a nonfiction book costs $20. The total cost of the order was $900.
How many nonfiction books did the librarian order? Show your work.

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 1, MP 5

Look back at the information on page 497 about Steve and Sally’s canoe trip.

a. How much time did Steve and Sally spend paddling?

b. In the Apply What You’ve Learned in Lesson 8-5, you found two expressions that
each represent the total amount of time Steve and Sally paddled during their trip,
in terms of their paddling speed x. Use one of those expressions and your answer to
part (a) to write an equation that you can solve for x.

c. Describe how to solve the equation you wrote in part (b) with a graphing calculator.

d. Solve the equation you wrote in part (b). Round your solutions to the nearest
hundredth. Check your answer by another method.

e. Interpret your solutions to the equation in terms of the problem situation.

548 Chapter 8  Rational Functions

Concept Byte Systems With CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Rational Equations
For Use With Lesson 8-6 Extends AM-ARFESI..D9.1121. AE-xRpElaI.i4n.1w1h yEtxhpelaxi-ncowohryditnhaetes of
txh-ecopooridnitnsawtehseorfetthheepgorainpthsswohfetrheetehqeugartaiopnhs yof=thf(ex)
aeqnudayti=ongs(xy)=inft(exr)saecntdayre=thge(xs)oinlutteirosnesctoaf rtehetheequation
fs(oxl)u=tiogn(sx)o.f.t.he equation f(x) = g(x) . . .

MP 1

You can solve systems with rational equations using some of same methods you used
with linear systems.

1 y = x
y = 3x − 1.
Follow each direction to solve the system • 1

x+1

1. Set the expressions for y equal to each other.

2. Solve for x.

3. Check your answer by substituting in the original system.

2 6

Follow each direction to solve the system • x − 2 = y .
y + 1 = x

4. Solve each equation for y.

5. Set the resulting expressions equal to each other.

6. Solve for x.

7. Check your answer by substituting in the original system.

Exercises

Solve each system.

8. • x2 - y + 3 = -2 9. • y = 1 3
4x y = x -
4
x - 2y = 3 x 2

y = x 2 - 2x - 2 y = x2 x +2 2 + 2
+ 3x +
10. • x2 + x - 6 11. •
y = x+3 y-3=x

12. Reasoning ​It is possible for the graph of a system of rational equations to include
a point of intersection that is an extraneous solution? Explain.

Concept Byte  Systems With Rational Equations 549

Concept Byte Rational Inequalities MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

For Use With Lesson 8-6 Extends AM-ARFESI..D9.1121. AE-xRpElaI.i4n.1w1h yEtxhpelain
xw-hcyoothrdeinxa-ctoeosrodfintahteeps ooifntthsewphoeinretsthweh gereaphs
technology tohfethgeraepqhusaotifotnhseyeq=ufa(txio) nasndy =y =f(xg)(ax)nd
yin=tergs(exc)tinatrertsheectsaorleuttihoenssolfuthioenesqoufatthioen
Consider the rational inequality 4 x x 6 3. fe(qxu) a=tigo(nx)f(.x.).= g(x) . . .
- MP 5

1  MATHEMATICAL

1. Enter y1 = 4 x x and y2 = 3 in your graphing calculator. Graph the functions PRACTICES
-
using the settings at the right. Use the calculator’s intersect feature to find WINDOW
Xmin=Ϫ6
where the two functions are equal. Use the graph to find the solution of the Xmax=12
x Xscl=1
inequality 4 - x 6 3. Ymin=Ϫ7
Ymax=5
2 Yscl=1
Xres=1

Using the functions you entered in Activity 1, set up the table as shown.

TABLE SETUP X Y1 Y2
TblStart = –1
∆Tbl=1 –1 –.2 3
0 0 3
Indpnt: Auto Ask 1 .33333 3
Depend: Auto Ask 2 1 3
3 3 3
4 ERROR 3
5 –5 3
 
X=−1

2. Scroll to x-values less than 3. Do they make the inequality true?

3. Scroll to x-values greater than 4. Do they make the inequality true?

4. What happens to the inequality when x = 3? When x = 4?

5. Change ΔTbl to 0.1. Investigate the inequality between x = 3and x = 4.

6. Make a conjecture about the solution of the inequality based on your results in
Steps 2–5.

3

Now use algebra to solve the inequality. You can multiply both sides of a rational

inequality by the same algebraic expression just as you have done with equations.

But you must keep in mind the properties of inequalities. Consider the first step,

multiplying each side by (4 - x).

4 x x 6 3
-

(4 - x) 4 x x 6 (4 - x) 3
-

Depending on whether the factor (4 - x) is positive or negative, there are two
possible solutions to the inequality.

550 Concept Byte  Rational Inequalities

7. First, consider the case where 4 - x 7 0, or x 6 4. Justify each step.
Hint: The solution must satisfy both x 6 3 and x 6 4.
x
4 - x 6 3

a. (4 - x) 4 x x 6 (4 - x) 3
-

b.   x 6 12 - 3x

c.   4x 6 12

d.   x63

8. Combine this result with the given condition x 6 4. What is the solution for
this case?

9. Now consider the case where 4 - x 6 0, or x 7 4. Justify each step.
Hint: The solution must satisfy both x 7 3 and x 7 4.
x
4 - x 6 3

a. (4 - x) 4 x x 7 (4 - x) 3
-

b.   x 7 12 - 3x

c.   4x 7 12

d.   x73

10. Combine this result with the given condition x 7 4. What is the solution for
this case?

11. Examine your solutions for Exercises 8 and 10. Now, write the solution of the
x
inequality 4 - x 6 3.

Exercises

For Exercises 12–17, solve each inequality graphically and algebraically.

12. x 2 1 6 x 13. x + 1 7 x + 52 14. (x - 2x + 3) 6 1
- x + 2)(x

15. 2x + 2 6 x + 1 16. x2 + 1 6 2x 17. x - 1 6 x + 3
x-1 x x - 2 x - 1

18. For Exercises 12–17, check your work by using a table to solve each inequality.

The equation d = rt relates the distance d you travel, the time t it takes to travel that

distance, and the rate r at which you travel. So the time it takes to travel a distance

d at a rate r is t = d . If you increase your rate by a to r + a, then it takes less time,
r
d d d
t = r + a. In fact, the time you save by going at the faster rate is T = r − r + a.

19. a. You normally take a 500-mi trip, averaging 45 mi/h. You want to increase the

rate so that you save at least an hour. Write an inequality that describes the

situation.
b. Solve your inequality from part (a).

Concept Byte  Rational Inequalities 551

8 Pull It All Together

RMANCPERFOE TASKCompleting the Performance Task

To solve these Look back at your results from the Apply What You’ve Learned sections in
problems, Lessons 8-2, 8-5, and 8-6. Use the work you did to complete the following.
you will pull
together 1. Solve the problem in the Task Description on page 497 by determining
concepts and how long it took Steve and Sally to paddle to Elk Island. Assume that
skills related Steve and Sally always paddle at the same speed. Round your answer to
to rational the nearest 5 minutes. Show all your work and explain each step of your
expressions, solution.
functions, and
equations. 2. Reflect  Choose one of the Mathematical Practices below and explain
how you applied it in your work on the Performance Task.

MP 1: Make sense of problems and persevere in solving them.

MP 2: Reason abstractly and quantitatively.

MP 4: Model with mathematics.

MP 5: Use appropriate tools strategically.

On Your Own

The next time Steve and Sally went on a canoe trip on Red River, the current
was flowing at a different rate. This time they left at 8:00 a.m. and returned
at 5:30 p.m. They paddled 15 mi downstream and then returned to their
starting point by paddling back upstream. They took two 45 min breaks
during the day.

Assuming that Steve and Sally paddle at a steady speed of 5 mi/h in still
water, what was the speed of Red River’s current during Steve and Sally’s
canoe trip?

552 Chapter 8  Pull It All Together

8 Chapter Review

Connecting and Answering the Essential Questions

1 Proportionality Inverse Variation  (Lesson 8-1) The Reciprocal Function Family

Quantities x and y are Are / and w inversely proportional? (Lesson 8-2)

inversely proportional only A = /w ᐉ A (a/)st=ret5/ch A
P = 2/ + 2w is 4
if growing x by the factor k

(k 7the1)famcteoarn1ks.shrinking y • for a constant of the
by
area—yes w graph of 2
• f or a constant bAy(/a)f=act1/or O 2 4ᐉ
of 5.
perimeter—no

2 Function Rational Functions and Their Graphs
A rational function may
have no asymptotes, one (Lesson 8-3) 4y Solving Equations Involving
horizontal or oblique FAosrymy p=toxt2e2sx-:2 9
asymptote, and any number horizontal: y = 2 x Rational Expressions  (Lessons 8-4,
of vertical asymptotes. Ϫ4 O 4
8-5, and 8-6)
3 Eaf (nqxd)u=giv(xxxa2) +-l=eaan2x,c-1xea≠, x{≠a,{a, 2x2
are equivalent. vertical: x = {3 x2 - 9 = x - 6 + 18 9
x - 3 x2 -

For y = 2x3 + 6x2 2x2 = (x - 6)(x + 3) + 18 9
x2 + 1 x2 - 9 (x - 3)(x + 3) x2 -

oblique: y = 2x + 6. 2x2 = x2 - 3x - 18 + 18

y = x4 + 5 has no asymptotes. x2 + 3x = 0
x2 + 1
x(x + 3) = 0

x = 0 ✓  or   x = - 3 ✗

Chapter Vocabulary

• branch (p. 508) • discontinuous graph • oblique asymptote (p. 524) • reciprocal function
• combined variation (p. 516) • point of discontinuity (p. 507)

(p. 501) • inverse variation (p. 498) (p. 516) • removable discontinuity
• complex fraction (p. 536) • joint variation (p. 501) • rational equation (p. 542) (p. 516)
• continuous graph (p. 516) • non-removable • rational expression (p. 527)
• rational function (p. 515) • simplest form (p. 527)
discontinuity (p. 516)

Choose the correct term to complete each sentence.

1. When the numerator and denominator of a rational expression are polynomials with
no common factors, the rational expression is in ? .

2. If a quantity varies directly with one quantity and inversely with another, it is a(n) ? .

3. A(n) ? has a fraction in its numerator, denominator, or both.

4. If a is a zero of the polynomial denominator of a rational function, the function has
a(n) ? at x = a.

5. A(n) ? of the graph of a rational function is one of the continuous pieces of its graph.

Chapter 8  Chapter Review 553

8-1  Inverse Variation

Quick Review Exercises

An equation in two variables of the form y = k or xy = k, 6. Suppose that x and y vary inversely, and x = 30
x when y = 2. Find y when x = 5.
where k ≠ 0, is an inverse variation with a constant of

variation k. Joint variation describes when one quantity Write a direct or inverse variation equation for each
relation.
varies directly with two or more other quantities.

Example 7. x 34 8 8. x 579
y 24 18 9 y 30 42 54
Suppose that x and y vary inversely, and x = 10 when
y = 15. Write a function that models the inverse Write the function that models each relationship. Find z
when x = 4 and y = 8.
variation. Find y when x = 6.
9. z varies jointly with x and y. When x = 2 and y = 2,
y = k z = 7.
x
10. z varies directly with x and inversely with y. When
15 = k , so k = 150. x = 5 and y = 2, z = 10.
10

The inverse variation is y = 15x0.

When x = 6, y = 150 = 25.
6

8-2  The Reciprocal Function Family

Quick Review Exercises

The graph of a reciprocal function has two parts called Graph each equation. Identify the x- and y-intercepts and
byr=ankxcbhyebs.uTnhietsghraoprihzoonftyal=lyxan-k db
+ c is a translation of the asymptotes of the graph.
c units vertically. It has
11. y = 1
a vertical asymptote at x = b and a horizontal asymptote x

at y = c. 12. y = -2
x2

Example 13. y = -x1 - 4

Graph the equation y = x 3 2 + 1. Identify the x- and 2
− +
y-intercepts and the asymptotes of the graph. 14. y = - 1
x 3

b = 2, so the vertical asymptote is x = 2. Write an equation for the translation of y = 4 that has the
x
c = 1, so the horizontal asymptote is y = 1. given asymptotes.

Translate y = 3 two units to the right and one unit up. 15. x = 0, y = 3
x 16. x = 2, y = 2

When y = 0, x = -1. 4y

The x-intercept is ( -1, 0). Ϫ8 Ϫ4 O 4 x 17. x = -3, y = -4
When x = 0, y = - 12. Ϫ4 18. x = 4, y = -3
The y-intercept is (0, - 21).

554 Chapter 8  Chapter Review

8-3  Rational Functions and Their Graphs

Quick Review P(x) Exercises
Q(x)
The rational function f (x) = has a point of Find any points of discontinuity for each rational
discontinuity for each real function. Sketch the graph. Describe any vertical or
zero of Q (x). horizontal asymptotes and any holes.

If P (x) and Q (x) have

• no common factors, then f (x) has a vertical asymptote 19. y = (x x-1 1)
when Q (x) = 0. + 2)(x -

• a common real zero a, then there is a hole or a vertical 20. y = x3 - 1
asymptote at x = a. x2 - 1

• degree of P (x) 6 degree of Q (x), then the graph of f (x) 21. y = 2x2 + 3
has a horizontal asymptote at y = 0. x2 + 2

• degree of P (x) = degree of Q (x), then there is a 22. The start-up cost of a company is $150,000. It
horizontal asymptote at y = ab, where a and b are the costs $.17 to manufacture each headset. Graph
coefficients of the terms of greatest degree in P (x) and the function that represents the average cost of a
headset. How many must be manufactured to result
Q (x), respectively. in a cost of less than $5 per headset?
• degree of P (x) 7 degree of Q (x), then there is no

horizontal asymptote.

Example

Find any points of discontinuity for the graph of the
2.5
rational function y = x + 7. Describe any vertical or

horizontal asymptotes and any holes.

There is a vertical asymptote at x = -7 and a horizontal
asymptote at y = 0.

8-4  Rational Expressions

Quick Review Exercises

A rational expression is in simplest form when its Simplify each rational expression. State any restrictions
numerator and denominator are polynomials that have on the variable.
no common factors.
x2 + 10x + 25
23. x2 + 9x + 20

Example # 24. x2 - 2x - 24 x2 - 1
x2 + 7x + 12 x-6
Simplify the rational expression. State any restrictions on

the variable. 4x2 - 2x 2x
x2 + 5x + 4 + 2x
#2x2 + 7x + 3 x2 − 16 25. , x2 + 1

x−4 x2 + 8x + 15

5 (2x 1 1)(x 1 3) ? (x 2 4)(x 1 4) 26. What is the ratio of the volume of a sphere to its
x24 (x 1 3)(x 1 5) surface area?

= (2x + 1)(x + 4) , x ≠ - 5, x ≠ - 3, and x ≠ 4
x+5

Chapter 8  Chapter Review 555

8-5  Adding and Subtracting Rational Expressions

Quick Review Exercises

To add or subtract rational expressions with different Simplify the sum or difference. State any restrictions on
denominators, write each expression with the LCD. A the variable.
fraction that has a fraction in its numerator or denominator
or in both is called a complex fraction. Sometimes you can 27. 3x 4 + x 6 2
simplify a complex fraction by multiplying the numerator x2 - +
and denominator by the LCD of all the rational expressions.
28. x2 1 1 - x2 2 3x
- +

Example 1 + 3 Simplify the complex fraction.
x
Simplify the complex fraction.  5 2
( ) #1 y + 4 2 - x
1 29. 3 - 1
x + 3 x + 3 xy x
5 + 4 xy
= 5 + 4 1
( ) #y y
# #= x+y
1 xy + 3 xy 30. 4
x xy + 4 xy
5# #y

= y + 3xy
5x + 4xy

8-6  Solving Rational Equations

Quick Review Exercises

Solving a rational equation often requires multiplying Solve each equation. Check your solutions.
each side by an algebraic expression. This may introduce
extraneous solutions—solutions that solve the derived 31. 1 = x 5 4
equation but not the original equation. Check all possible x -
solutions in the original equation.
32. x 2 3 - 1 = -6 3)
+ x x(x +

Example 33. 1 + x = 18
2 6 x
Solve the equation. Check your solution.

1 − 2 = 1 34. You travel 10 mi on your bicycle in the same amount
2x 5x 2 of time it takes your friend to travel 8 mi on his
bicycle. If your friend rides his bike 2 mi/h slower
( ) ( )10x1 - 2 1 than you ride your bike, find the rate at which each
2x 5x = 10x 2 of you is traveling.

5 - 4 = 5x

x = 1
5

Check  1 - 2 = 5 - 2 = 1 ✔
2 2
2 1 15 2 5 1 15 2

556 Chapter 8  Chapter Review

8 Chapter Test M athX

OLMathXL® for School
R SCHO Go to PowerAlgebra.com
L®

FO

Do you know HOW? Simplify each rational expression. State any
Write a function that models each variation. restrictions on the variable.

1. x = 2 when y = -8, and y varies inversely with x. 16. x2 + 7x + 12
x2 - 9

2. x = 0.2 and y = 3 when z = 2, and z varies jointly 17. (x + 3)(2x - 1) , (-x - 3)(2x + 1)
with x and y. x(x + 4) x

3. x = 31, y = 51, and r = 3 when z = 21, and z varies r 2 18. x2 x2 - 1 3 - x + 1
directly with x and inversely with the product of + 2x - x + 3

and y. x(x + 4) x-1
x-2 x2 - 4
19. +

Is the relationship between the values in each table a Solve each equation. Check your solutions.
direct variation, an inverse variation, or neither? Write
equations to model any direct or inverse variations.

4. 5. x 20. x = x + 1 21. x 3 1 = 3x 4 2
4 2 4 - +
x y 8 y
3 6 16 32 22. x 3x 1 = 0 23. x 3 1 = x2 1 1
5 8 32 16 + + -
7 10
9 12 8 24. 1 + 1 = 6 25. 1 + x x 2 = 1
4 x 3 x2 x +

Write and graph an equation of the translation of y = 7 26. Your neighbor can seal your driveway in 4 hours.
x Working together, you and your neighbor can seal
that has the given asymptotes. it in 2.3 hours. How long would it take you to seal it
working alone?

6. x = 1; y = 2 7. x = -3; y = -2

For each rational function, identify any holes or Do you UNDERSTAND?
horizontal or vertical asymptotes of the graph. 27. Vocabulary  ​Describe a situation that represents an

8. y = x + 11 9. y = x + 3 inverse variation.
x - x + 3
28. Compare and Contrast  ​How is simplifying rational
10. y = (x x-2 2) 11. y = 2x2 expressions similar to simplifying fractions? How is
+ 1)(x - x2 - 4x it different?

12. y = x 1 2 - 3 13. y = x2 + 5 29. Writing  ​When does a discontinuity result in a vertical
+ x- 5 asymptote? When does it result in a hole in the graph?

Simplify each complex fraction. 30. Open-Ended  W​ rite a function whose graph has a
hole, a vertical asymptote, and a horizontal asymptote.
2 3 - 3
x - x 31. Reasoning  ​State any restrictions on the variable in
14. 1 1 15. 1 1 the complex fraction.  x - 3
- 2 x
y x+4
x 2 x- 1

Chapter 8  Chapter Test 557

8 Common Core Cumulative ASSESSMENT
Standards Review

Some problems ask you to What is the approximate lateral area of the TIP 2
find the lateral area or the cone shown below?
(total) surface area of a three- Use the information
dimensional figure. Read the x cm from the diagram for the
sample question at the right. 2 values you need in the
Then follow the tips to answer formula.
the question.
4 cm Think It Through
TIP 1 − x
x2 3x radius: r = 2
Use the formula for the
lateral area of a cone: 3 3 slant height: / = x2 4 3x
S = PrO. + - S = pr/ -
x 3 x 3

6 6 ( ) ( )= px
x - 3 x + 3 2 4
x2 - 3x
2p
= x-3

Since p ≈ 3, the correct

answer is B.

VLVeooscsacoabnubluarlayry Builder Selected Response

As you solve test items, you must understand Read each question. Then write the letter of the correct
the meanings of mathematical terms. Match each answer on your paper.
term with its mathematical meaning.
1. Which expression equals 5x - x2 4x 6?
x2 - 9 + 5x +
A. joint variation I. a point where the graph
7x
B. branch of a function breaks into (x - 3)(x + 3)(x + 2)
branches
x2 -
C. point of II. each piece of a (x - 3)(x + 2x + 2)
discontinuity discontinuous graph 3)(x

D. inverse variation I II. a relation represented by an (x - x2 + 22x + 2)
3)(x + 3)(x
E. reciprocal
function equation of the form y = k 9x2 - 2x
x 3)(x + 3)(x
or xy = k, where k ≠ 0 (x - +
2)

IV. a function that can be written 2. If x is a real number, for what values of x is the
2x - 8 x2 - 4x
in the form f (x) = a + k, equation 4x-1 = 2 true?
where a ≠ 0 x - h

all values of x

V. one variable varies directly some values of x
with two or more other
variables no values of x
impossible to determine

558 Chapter 8  Common Core Cumulative Standards Review

3. Which expression represents the solution to 4x = 13? 8. Which is the simplest form of the expression?
4 218x4 - 3 272x4
log 13
log 4
- 6x2 22 -6
log4 + log13 - 6x2 none of the above

log 4 9. If g (x) = x2 - 4 and h (1xhg)2=(x4)x? - 6, which
log 13 expression is equal to

log134

4. If f (x) = x2 and g (x) = x - 1, which statement is 4x - 6
x2 - 4
#true?
x2 - 2
f (x) g (x) = 2x3 - 1 4x - 3

f (x) - g (x) = x - 1 x2 - 4 - (4x - 6)
f (x) - g (x) = x2 - x + 1
f (x) + g (x) = x3 - 1 (x + 2) (x - 2)
2(2x - 3)

5. Which expression is a simpler form of the complex 1 0. If i = 1-1, what is the value of -i4?

1 + 3 i 1

fraction x 2 y? -i -1

3xy xy 11. The graph below shows the transformation of the
2 function f (x) = (x + 3)2. Which quadratic function
3 models this graph?
2
3x + y 3x + y
2xy 2

6. Which is the first incorrect step in simplifying log9 243? y

Step 1: log 9 243 = x 6

Step 2: 9x = 243 4

Step 3: x = 243 , 9 2

Step 4: = 27 x
Step 1 O 24

Step 2 f (x) = (x - 1)2 f (x) = (x - 3)2 - 2
f (x) = 2(x - 3)2 f (x) = (x - 3)2 + 2
Step 3

Step 4 1 2. What is the solution of 3e2x + 1 = 5?

7. Which is/are the solution(s) of the equation x ≈ 0.131 x ≈ 0.187
12x + 2 = 2x - 4?
x ≈ 0.151 x ≈ 0.144
x = 3.5 and x = 1

x = 3.5 and x = -1

x = 3.5

x = 1

Chapter 8  Common Core Cumulative Standards Review 559

Constructed Response 2 4. Explain how you know that the graph of
f(x) = -x4 - 3x + 7 does not have up and down
13. Solve for x: 2x 5 2 = 15 1. end behavior.
- x2 -
25. Graph y = 3x+2 - 5.
1 4. Solve for x: log(x - 3) = 3.
2 6. How can the solution of 4x = 13 be written as a
1 5. What is the value of the x-coordinate of the logarithm?

solution of the system of equations? ( )2 7. If g(x) = x2 - 4 and h(x) = 4x - 6, write an

e 2x + y = 6 expression equivalent to g (x) in factored form.
y-3=x h

16. What is the remainder when x4 - 3x2 + 7x + 3 is Extended Response
divided by x - 2?
28. Explain how to find an equation for the translation of
1 7. The product of three consecutive even integers is 3
-2688. What is the value of the largest integer? y = x that has asymptotes at x = - 13 and y = 5.

18. What is the smallest zero of f (x) = 2x5 - 4x2 + 3x + 7? 2 9. What is the inverse of y = x2 + 15? Is the inverse a
Round your answer to the nearest hundredth.
function? Explain.

19. The matrix below represents a linear system of 30. What are the solutions of 3x2 + 4x + 2 = 0? Show
your work.

equations. What is the y-coefficient of the first equation 3 1. You are given the equation y = 2 0 x - 1 0 + 3.

of the system? a. Without graphing, what are the vertex and axis of

c 31 -1 ` 5 symmetry of this function?
2 -1d b. Describe the transformations from the parent

2 0. What is the sum of the zeros of the polynomial function function y = 0 x 0 .
y = x2 - 4y - 5?
c. Sketch the graph.
21. Write the radical expression 250x 5y 3z in simplest form.
What is the constant value under the radical sign? 32. Without solving, explain how you know that the

2 2. If p(x) = 3x4 - 2x3 + x2 - 1 and g (x) = x - 2, what equation 13x - 4 + 1x + 3 = -2 has no real
is the remainder of gp((xx))? Show your work. solutions.

23. Solve the equation 2g 2 - c = 3x for g. Show your
d
work.

560 Chapter 8  Common Core Cumulative Standards Review