Angel, Elementary and Intermediate Algebra, 3ed 1 Rational Expressions and Equations Chapter 7

Angel, Elementary and Intermediate Algebra, 3ed 1

Rational Expressions

and Equations

Chapter 7


§ 7.1

Simplifying Rational

Expressions




A rational expression is an expression of the form p/q where p and q are polynomials and q 0.

Examples:

Whenever a rational expression contains a variable in the denominator, assume that the values that make the denominator 0 are excluded.


Signs of a FractionThree signs are associated with any fraction: the sign of the numerator, the sign of the denominator, and the sign of the fraction.

Changing any two of the three signs of a fraction does not change the value of the fraction

- a+ b

+

sign of the numerator

sign of the denominator

sign of the fraction

- a b

a -b

= a b

-=


Simplifying

A rational expression is simplified or reduced to lowest terms when the numerator and denominator have no common factors other than 1. Examples:



Simplifying Rational Expressions

1. Factor both the numerator and denominator as completely as possible.

2. Divide out any factors common to both the numerator and denominator.


Factoring a Negative 1

Remember that when –1 is factored from a polynomial, the sign of each term in the polynomial changes.

Example: – 2x + 5 = – 1(2x – 5) = –(2x – 5)




§ 7.2

Multiplication and Division of



Multiplying Fractions

0d and 0b,db

ca

d

c

b

a

Multiply . - 521

·34

- 521

·34

=- 521

·34

1

7

- 57

·14

= =- 528


Multiplying Rational Expressions

1. Factor all numerators and denominators completely.

2. Divide out common factors.3. Multiply numerators together and

multiply denominators together.

Multiply .yx

22z

11z

y18x-52

32

52

32

yx

22z

11z

y18x-4

2

y

36z

52

32

yx

22z

11z

y18x-



Dividing Two Fractions

0c and 0d , 0b ,bc

ad

c

d

b

a

d

c

b

a

Divide . - 2 9

59

=- 2 9

59

- 2 9

·95

1- 2 5

=

1

- 2 9

·95

=


Dividing Rational Expressions

Invert the divisor (the second fraction) and multiply

Divide .3017-x

1

127xx

122

1

3017-x

187xx

1

3017-x

1

187xx

1 2

222

1

15)2)(x(x

2)9)(x(x

1

9)(x

15)(x



Factor the x-box way

Example: Factor 3x2 -13x -10

-13x

(3x2)(-10)=

-30x2

-15x 2x

-10

-15x

2x

3x2

x -5

3x

+2

3x2 -13x -10 = (x-5)(3x+2)



Example: Factor 2x2 +3x -9

3x

(2x2)(-9)=

-18x2

6x -3x

-9

6x

-3x

2x2

x +3

2x

-3

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



Example: Factor 3x2 -2x -5

-2x

(3x2)(-5)=

-15x2

-5x 3x

-5

-5x

3x

3x2

3x -5

x

+1

3x2 -2x -5=(3x-5)(x+1)



§ 7.3

Addition and Subtraction of

Rational Expressions with a

Common Denominator


Adding/Subtracting Fractions

0c ,c

ba

c

b

c

a 0c ,

c

ba

c

b

c

a

712

= 512

212

+

Add . 512

212

+


Common Denominators

1. Add or subtract the numerators.2. Place the sum or difference of the

numerators found in step 1 over the common denominator.

3. Simplify the fraction if possible.

Subtract .5

6

5

7-2x

5

13-2x

5

6-7-2x

5

6

5

7-2x


Common Denominators

a.) Add .12ww

4-2w-

12ww

53w22

Example:

12ww

4-2w-53w

12ww

4-2w-

12ww

53w222

1)(w

1

1)(w

1w2

12ww

4-2w-53w2


Common Denominators

b.) Subtract

.649x

29x-x

649x

54x2

2

2

2

649x

29)x-(x-54x

649x

29x-x

649x

54x2

22

2

2

2

2

649x

24x3x

649x

29xx-54x2

2

2

22

8)(3x

3)(x

8)8)(3x(3x

8)3)(3x(x

Example:


Least Common Denominator

1. Factor each denominator completely. Any factors used more than once should be expressed as powers.

2. List all different factors that appear in any of the denominators. When the same factor appears in more than one denominator, write that factor with the highest power that appears.

3. The least common denominator (LCD) is the product of all the factors listed in step 2.


Least Common Denominator

a.)5-x

6

52x

3-x

Find the LCD:

The LCD is (2x + 5)(x – 5).

x

x

xx

2-5x 2

2

b.) The LCD is x(x + 1).x

x

1)x(x

2-5x 2

245 9wz

2

z4w

3c.) The LCD is 36w5z4

.


§ 7.4

Addition and Subtraction of



Unlike Denominators

1. Determine the LCD.2. Rewrite each fraction as an

equivalent fraction with the LCD.3. Add or subtract the numerators

while maintaining the LCD.4. When possible, factor the

remaining numerator and simplify the fraction.


Unlike Denominators

a.)w

5

2w

3

2w

2w

w

5

w

w

2w

3

The LCD is w(w+2).

2)w(w

2)5(w

2)w(w

3w

2)w(w

105w

2)w(w

3w

2)w(w

108w

answers. acceptable also are and 2ww

108w

2)w(w

5)2(4w2

Example:


Unlike Denominators

b.)3x

1

4-4x

x The LCD is 12x(x – 1).

3x

1

1)-4(x

x

1)-4(x

1)-4(x

3x

1

3x

3x

1)-4(x

x

1)-12x(x

44x3x

1)-12x(x

1)-4(x

1)-12x(x

3x 22

This cannot be factored any further.

Example:


Unlike Denominators

c.)5-x

2

10-3x-x

3x2

The LCD is

(x – 5)(x+2).5-x

2

2)5)(x-(x

3x

2x

2x

5-x

2

2)5)(x-(x

3x

2)5)(x-(x

2)2(x

2)5)(x-(x

3x

2)5)(x-(x

42x

2)5)(x-(x

3x

2)5)(x-(x

4)(2x-3x

2)5)(x-(x

42x3x

2)5)(x-(x

1x

Example:


§ 7.5

Complex Fractions


Simplifying Complex Fractions

A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator.

454

3xx

3x

ba9-a

ba

Example:


Simplify by Combining Like Terms

1. Add or subtract the fraction in both the numerator and denominator of the complex fraction to obtain single fractions in both the numerator and the denominator.

2. Invert the denominator of the complex fraction and multiply the numerator by it.

3. Simplify further if possible.


4

b

b

12a 2

3

4bb

12a

2

3a)

23 b

4

b

12a

5b

48a

yx

y-xx

2b

) y

x

y-x

x 2

2x

y

y-x

x

y)-(xx

xy2

Simplify by Combining Like Terms

y)-x(x

y

Simplify:


Simplify by Multiplying

1. Find the LCD of all the denominators

appearing in the complex fraction.

2. Multiply both the numerator and the

denominator of the complex fraction by

the LCD.

3. Simplify further if possible.


Simplify by Multiplying

4bb

12a

2

3a)

LCD is 4b3.

4b3

4b

b12a

2

3

4b3

44bb

48ab

5

3

3

5b

48a

yx

y-xx

2b

)LCD is y(x-y).

y(x-y)

y(x-y)yx

y-xx

2

yy)-y(xx

y-xy)-xy(x

2y)-(xx

xy2

y)-x(x

y

Simplify:


§ 7.6

Solving Rational Equations


Complex Fractions

A rational equation is one that contains one or more rational (fractional) expressions.

8x5

3x

2

1 5

3x

3

Example:


Solving Rational Equations

1. Find the LCD of all fractions in the equation.

2. Multiply both sides of the equation by the LCD. (Every term will be multiplied by the LCD.)

3. Remove any parentheses and combine like terms on each side of the equation.

4. Solve the equation.

5. Check the solution in the original equation.


Integer Denominators

46

x

5

xa

)The LCD is 30. 304

6

x

5

x30

1206

30x

5

30x 1205x6x 120x

CHECK: 46

120

5

120 42024

Solve the equation:


Variable Denominators

The LCD is 2z.Solve: 2

11

z

35

CHECK:

Whenever there is a variable in the denominator, it is necessary to check your answer in the original equation. If the answer obtained makes the denominator zero, that value is NOT a solution to the equation.

2

112z

z

352z

2

22z

z

6z10z 11z610z z6

2

11

z

35

2

11

2

15

2

11

6

35


Variable Denominators

The LCD is 2(x-3).Solve:

3-x

3

2

3

3-x

x

CHECK:

3-x

33)2(x

2

3

3-x

x3)2(x

3-x

3)-6(x

2

3)-6(x

3-x

3)-2x(x 63)-3(x2x

693x2x 155x 3x

3-3

3

2

3

3-3

3

0

3

2

3

0

3

No solution


§ 7.7

Rational Equations:

Applications & Problem Solving


Work Problems

Problems in which two or more people or machines work together to complete a certain task are referred to as work problems.

part of task done by first machine

part of task done by second machine

1 (one whole task

completed)+ =


Work Problems

At the NCNB Savings Bank it takes a computer 4 hours to process and print payroll checks. When a second computer is used and the two computers work together, the checks can be printed in 3 hours. How long would it take the second computer by itself to process and print the payroll checks?

Example:

Continued.


Work Problems

Machine

Rate of Work Time

Part of Task

(rate x time)

1st

2nd

A table helps to keep information

organized.

Example continued:

Continued.


Work Problems

Machine

Rate of Work Time

Part of Task

(rate x time)

1st

2nd

4

1

x

1

4

3

x

3

1x

3

4

3 One job done

3

3

Solve the equation.

Example continued:

Continued.


Work Problems

1x

3

4

3 The LCD is 4x.

1

x

3

4

34x 4x123x x12

It would take the second computer 12 hours by itself.

CHECK: 112

3

4

3 1

12

3

12

9

Example continued:

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Angel, Elementary and Intermediate Algebra, 3ed 1 Rational Expressions and Equations Chapter 7