section 6.1 rational expressions. objectives determine the values that make a rational expression...

Section 6.1

Rational Expressions

OBJECTIVES

Determine the values that make a rational expression undefined.

A

OBJECTIVES

Build fractions.B

Reduce (simplify) a rational expression to lowest terms.

C

RULESAvoiding Zero DenominatorsThe variables in a rational expression must not be replaced by numbers that make the denominator zero.

RULES

= • 0, 0

A A C B CB B C

Fundamental Rule of Rational Expressions

PROCEDUREReducing Rational Expressions to Lowest Terms

1. Write the numerator and denominator of the expression in factored form.


2. Find the factors that are common to the numerator and denominator.


aa = 1.

3. Replace the quotient of the common factors by the number 1 since


4. Rewrite the expression in simplified form.

RULE

a – bb – a

= – 1

Quotient of Additive Inverses

Section 6.1Exercise #1

Chapter 6Rational Expressions

Write 3x

7y with a denominator of 21y3

3x7y

= ?21y 3

Note: 21y 3 =

• 3y 2

7y

3x7y

=

3x • 3y 2

7y • 3y 2 =

9xy 2

21y 3

Reduce to lowest terms.

– 6(x2 – y2 )3(x – y )

=

– 1 • 6 (x + y ) (x – y )3 (x – y )

1

1

2

1

= – 2(x + y )

Factor

Simplify

Determine the values for which the expression is undefined and simplify.

=

– 1 • xx (1 + x )

Factor

=

– 11 + x

Simplify 1

1

NOTE: Undefined if x + x2 = 0

– xx + x 2

x (1 + x ) = 0

x = 0 or 1 + x = 0

x = 0 or x = – 1

Undefined if x = 0 or x = – 1

Determine the values for which the expression – x

x + x2is undefined and simplify.

NOTE: Undefined if x + x2 = 0

Section 6.2


OBJECTIVES

Multiply two rational expressions.

A

Divide one rational expression by another.

B

RULEMultiplying Rational Expressions

• = 0, D 0

A C AC BB D BD

PROCEDUREMultiplying Rational Expressions

1. Reduce each expression if possible.

2. Multiply the numerators to obtain the new numerator.

PROCEDUREMultiplying Rational Expressions

3. Multiply denominators to obtain new denominator.

4. Reduce if possible.

RULE

Dividing Rational Expressions

÷ = • , and D 0

A C A D B CB D B C

Perform the indicated operation and simplify.

Multiply (x – 2) •

x + 3

x 2 – 4

=

(x – 2)

1 •

(x + 3)

(x + 2) (x 2)

=

x + 3x + 2

Factor

Simplify 1

1


Divide

x + 3x – 3

÷ x 2 – 93 – x

=

x + 3

x – 3 •

3 – x

x 2 – 9

Invert and multiply

=

– 1x – 3

or 1

3 – x

=

(x + 3)(x – 3)

• – 1 (x – 3)

(x + 3) (x – 3)

1

1 1

1

Factor

Section 6.3


OBJECTIVES

Add and subtract rational expressions with the same denominator.

A

OBJECTIVES

Add and subtract rational expressions with different denominators.

B

Solve an application.C

PROCEDUREAdding (or Subtracting) Fractions with Different Denominators.

1. Find the LCD.


2. Write all fractions as equivalent ones with LCD as the denominator.


3. Add or subtract numerators, keep denominators.


4. Reduce if possible.


Add

2x + 1

+ 1

x – 1

=

2(x – 1)

(x + 1)(x – 1) +

1 (x + 1)

(x + 1)(x – 1)

LCD = (x + 1)(x – 1)

Rewrite with LCD

=

2x – 2 + x + 1(x + 1)(x – 1)

=

2x – 2 + x + 1(x + 1)(x – 1)

=

3x – 1(x + 1)(x – 1)

or 3x – 1

x2 – 1

Simplify


Add

2x + 1

+ 1

x – 1


Subtract

x + 1

x2 + x – 2 –

x + 2

x 2 – 1

=

(x + 1)

(x + 2)(x – 1) –

(x + 2)

(x + 1)(x – 1)

LCD = (x + 2)(x – 1)(x + 1)

=

(x + 1)(x + 1)

(x + 2)(x – 1)(x + 1) –

(x + 2)(x + 2)

(x – 1)(x + 1)(x + 2)

=

(x2 + 2x + 1) – (x2 + 4x + 4)

(x + 2)(x – 1)(x + 1)

Remove parenthesis

=

x2 + 2x + 1 – x2 – 4x – 4(x + 2)(x – 1)(x + 1)

Rewrite with LCD

=

x2 + 2x + 1 – x2 – 4x – 4(x + 2)(x – 1)(x + 1)

Collect like terms

=

– 2x – 3(x + 2)(x – 1)(x + 1)

Section 6.4


OBJECTIVE

Simplify a complex fraction using one of two methods.

A

DEFINITION

A fraction with one or more fractions in its numerator, denominator or both.

Complex Fraction

PROCEDURESimplifying Complex Fractions

1. Multiply numerator and denominator by the LCD of the fractions involved, or

PROCEDURESimplifying Complex Fractions

2. Perform operations indicated in numerator and denominator, then divide simplified numerator by simplified denominator.


LCD = 12x

Simplify

1x

– 23x

34x

+ 12x

=

1x

– 23x

• 12x

34x

+ 12x

• 12x

=

1x • 12x

– 2

3x • 12x

34x • 12x

+ 1

2x • 12x

=

1 • 12 – 2 • 43 • 3 + 1 • 6

=

12 – 89 + 6

=

415

4

3 6

Section 6.5


OBJECTIVES

Solve equations that contain rational expressions.

A

OBJECTIVES

Solve a rational equation for a specified variable.

B

LEAST COMMON MULTIPLEMultiplying each side of the equation

by L is equivalent to multiplying each term by L.

ab

+ cd = e

f

Solve:

2 + 4x – 3

= 24

x 2 – 9 LCD = (x + 3)(x – 3)

2(x+3)(x – 3)+

4(x+3)(x – 3)x – 3

= 24(x+3)(x – 3)

(x+3)(x – 3)

2(x + 3)(x – 3) + 4(x + 3) = 24

2(x 2 – 9) + 4x + 12 = 24

2x 2 – 18 + 4x + 12 = 24

2x 2 – 18 + 4x + 12 = 24

2x 2 + 4x – 6 = 24

2x 2 + 4x – 30 = 0

x 2 + 2x – 15 = 0

LCD = (x + 3)(x – 3)

Solve:

2 + 4x – 3

= 24

x 2 – 9

x 2 + 2x – 15 = 0

(x+5)(x – 3) = 0

x+5 = 0 or x – 3 = 0

x = – 5 or x = 3

But x = 3 makes both denominators 0.

So, 3. The only solution is = – 5.x x

LCD = (x + 3)(x – 3)

Solve:

2 + 4x – 3

= 24

x 2 – 9

Solve for d1 .

D(1 + d)n =

d1(1 – dn )(1 + d)n

(1 + d)n

D(1 + d)n = d

1(1 + dn )

D(1 + d)n

(1 + dn ) = d

1

D =

d1(1 – dn )

(1 + d)n

Section 6.6


OBJECTIVES

Solve proportions.A

Solve applications.B

RULECross Products

If a

b = c

d, then

ad = bc

A car travels 150 miles on 9 gallons of gas. How many gallons will it need to travel 400 miles?

Let g = number of gallons needed to travel 400 miles.

9150

= g

400

ratio:

gallonsmiles

in each fraction

Translate

A car travels 150 miles on 9 gallons of gas. How many gallons will it need to travel 400 miles?

9150

= g

400 Use Algebra

9 • 400 = 150 • g

3600 = 150g

24 = g

24 gallons are needed to drive 400 miles.

A woman can paint a house in 5 hours. Another one cando it in 8 hours. How long would it take to paint the house if both women work together?

Let h = time it will take if they work together.

h5

+ h8

= 1 LCD = 40

8h + 5h = 40

13h = 40

A woman can paint a house in 5 hours. Another one cando it in 8 hours. How long would it take to paint the house if both women work together?

13h = 40

h = 3 1

3

It will take them 3 13

hours

if they work together.

A boat can travel 10 miles against a current in the same time it takes to travel 30 miles with the current. If the speed of thecurrent is 8 miles per hour, what is the speed of the boat in still water?

Let s = speed of the boat in still waters + 8 = speed with the currents – 8 = speed against the current

10s – 8

= 30

s + 8

10s – 8

= 30

s + 8

30(s – 8) = 10(s + 8)

30s – 240 = 10s + 80


30s – 240 = 10s + 80

20s = 320

s = 16

Boat's speed is 16 mi/hr in still water.


Find the unknown given similar triangles.

X

Y 5

Z

y

B

A

C

4

12

a.

y12

= 54

4y = 5 • 12

4y = 60

y = 15

Find the unknown given similar triangles. Q

R

P

r

10 A

B

C 7

5

b.

r7

= 105

5r = 7 • 10

5r = 70

r = 14

section 6.1 rational expressions. objectives determine the values that make a rational expression...

Documents

b slide

c slide

complex fraction slide

rational expression

complex fractions

different denominators

lowest terms

new denominator