Simplifying Rational

ExpressionsLesson 11-1

Rational Expression0Any fraction that has a variable in the numerator or the

denominator.

An extended value of a rational expression is when the denominator is undefined or equals zero. In this case, x can not be -6.

Therefore, x = -6 is a extended value.

Problem 10What is the simplified version of ? State any extended values. Find what the numerator and the denominator have in common. We want to cancel terms.

=

x – 1 can be cancelled since it appears in both the numerator and the denominator and it is being MULTIPLIED by a number.

Solution:

Got it? 1, a ≠ 0

, d ≠ -2

, n ≠

; none

Problem 20What is the simplified form of ? State any excluded

values.

= =

Look at the second part…what values of x would make the denominator zero?

Those are the excluded values. x 2 and x -3

Got it? 2 Simplify and give the excluded values.

, x ≠ -2, x ≠ 4

, x ≠ 1

, z ≠ -2, z ≠ -

, c ≠ -3, x ≠ -2

Opposite Numerators and Denominators

The numerator and denominator of are opposites.

To simplify the expression, we can factor out -1 from (3 – x) and can rewrite the denominator as -1(-3 + x) or -1(x – 3).

We can now simplify .

Problem 3Simplify and state any excluded values.

=

Set aside since that is fully simplified.

What’s left is .

Problem 3 can be simplified even further.

Because they are opposites, we can factor a -1 in the numerator. (HINT: factor -1 from the part of the fraction that makes the

variable negative)

= = -1

Now add the first part.

Problem 3Finding excluded values:

Take a look at the original problem .

What number can x NOT be? 7x – 14 = 0

7x = 14x = 2

x can not be 2.

Got it? 3-1, x ≠ -2.5

-y – 4 , y ≠ 4

- , d ≠ - , d ≠

- , z ≠ - 1, z ≠ 1

Problem 4A square has a side length of 6x + 2. A rectangle with a width of 3x + 1 has the same area as the square. What is the length of the rectangle?

(6x + 2)(6x + 2) = L (3x + 1)

L =

L =

L = 4(3x + 1)L = 12x + 4

Complete #30 as a class.

Multiplying and Dividing Rational ExpressionsLesson 11-2

Key Concept

x =

÷ =

Problem 1: What is the product?a. ∙

, where a 0

Problem 1: What is the product?b. ∙

where x 0, x -3

Got it? Find the product. a. ∙

b. ∙

, y ≠ 0

, x ≠ 2, x ≠ 3

Problem 2: Using FactoringWhat is the product of ∙ ?

Factor what you can.

∙ =

= 2

Got it? Find the productFind the product of ∙

=

3x(x + 1)

Problem 3: Multiplying by a Polynomial

What is the product of and (m2 + m – 6)?

Got it? 3 Multiply these expressions

∙ (6x2 – 13x + 6)

(x – 7)(3x – 2)

Dividing Rational Expressions

Step 1: Change the division sign to a multiplication sign.

Step 2: Take the reciprocal of the second part.

Step 3: Multiply and simplify.

Problem 4: Dividing Expressions ÷

•

•

•

Problem 4: Got it? ÷

Problem 5: Dividing Expressions ÷ (x2 – 3x – 4)

•

•

•

Problem 5: Got it? ÷ (z - 1)

Problem 6: Complex Expressions

÷

•

•

Problem 6: Got it?

Dividing Polynomials

Lesson 11-3

Problem 1: Dividing by a MonomialWhat is (9x3 – 6x2 + 15x) ÷ 3x2?

=

Cross off 3x in numerator and denominator.

Distribute the x to each term. - + = 3x – 2 +

Problem 1: Got it?What is (4d3 + 10d2 + 3d) ÷ 2d2?

2d + 5 +

Problem 2: Dividing by a Binomial

Compute 22 ÷ 5 using long division.

You will have a remainder.

Now try, (3d2 – 4d + 13) ÷ (d + 3) on the board.

Problem 2: Got it?

(2m2 – m + 3) ÷ (m + 1)

Answer: 2m – 3

Problem 3: Dividing with a Zero Coefficient

The width w of a rectangle is 3z – 1. The area A of the rectangle is 18z3 – 8z + 2. What is an expression for the length?

Step 1: We need to represent all of the degrees of z.

18z3 – 8z + 2 = 18z3 + 0z2 – 8z + 2

Use 18z3 + 0z2 – 8z + 2 when dividing.

Complete on the board.

Problem 3: Got it?

Divide:

(h3 – 4h + 12) ÷ (h + 3)

Answer: h2 – 3h + 5 -

Problem 4: Reordering Terms before Dividing

What is (-10x – 1 + 4x2) ÷ (-3 + 2x)

Step 1: We need to reorder them from greatest to smallest degree.

(-10x – 1 + 4x2) = (4x2 – 10x – 1)(-3 + 2x) = (2x – 3)

Now Divide.

Complete on the board.

Problem 4: Got it?

Divide:

(21a + 2 + 18a2) ÷ (5 + 6a)

Answer: 3a + 1 -

Adding and Subtracting Rational ExpressionsLesson 11-4

Review of adding and subtracting fractions:

Remember?

Like Denominators: add the numerator and keep the denominator the same.

Difference Denominators: find the least common multiple and change the denominators so they look the same. Then add the numerators.

Problem 1 and 2: Adding and Subtracting with Like Denominators

A. + + =

B. - = =

= = =

Problem 1 and 2 Got it?Add or subtract.

A. + =

B. - =

9𝑎3𝑎−1

3𝑛−45𝑛−2

Problem 3: Adding with Unlike Denominators

What is the sum + ?

Step 1: Find the lowest common multiple of the denominators.6x = 2 ∙ 3 ∙ x 2x2 = 2 ∙ x ∙ x LCM = 2 ∙ 3 ∙ x ∙ x = 6x2

Step 2: Rewrite each rational expression using the LCD and add.

+ = + =

Problem 3: Got it?

What is the sum + ?

.

Problem 4: Subtracting with Unlike Denominators

What is the difference - ?

Step 1: Find the LCD. Since there are no common factors the lowest common denominator is (d – 1)(d + 2).

Step 2: Rewrite each rational expression using the LCD and add.

- = =

Problem 4: Got it?

What is the sum - ?

.

Solving Rational EquationsLesson 11-5

Rational Equation

Rational = fractionEquation = has an equal sign

Example:

Multiply both sides by the LCD, 9.

6x = 12x = 2

Problem 10What is the solution of ?

The LCD is 12x.

5x – 6 = 45x = 10

x = 2

Got it? 1

x = -3

x = 5.29 or

Problem 2What are the solutions of 1 - ?

Find the LCD, which is x2. (x2)1 - (x2)

x2 – x = 12x2 – x – 12 = 0

(x + 3)(x – 4) = 0x = -3, 4

Got it? 2

y = -1.5, 1.5

d = -7, -1

Problem 3Amy can paint a loft apartment in 7 hours. Jeff can paint the same size loft apartment in 9 hours. If they work together, how long will it take them to paint a third loft apartment of the same size?

To solve a work problem, put each rate under 1 unit. Amy can paint in 7 hours, so she can paint of the room in an hour.

Jeremy can paint in 9 hours, so he can paint of the room in an hour.

Add the rates together and they should total , or the fraction of the loft painted in 1 hour.

Problem 3Amy can paint a loft apartment in 7 hours. Jeremy the same size loft apartment in 9 hours. If they work together, how long will it take them to paint a third loft apartment of the same size?

LCD is 63t.

9t + 7t = 6316t = 63

t = 3.9375 or 3

Got it? 3One hose can fill a poll in 12 hours and a second hose can fill the same pool in 8 hours. How long will it take for both hoses to fill the pool together?

LCD is 24t.

2t + 3t = 245t = 24

t = 4.8 or

Problem 4What is the solution of ?

These two ratios equal each other, so we can cross multiply.

4(x + 1) = 3(x + 2)4x + 4 = 3x + 6

x = 2

Got it? 4a. What is the solution of ?

b = -8

b. What is the solution of ?

c = -3, 7

Problem 5Sometimes the solution we find turns out to be false…here is an example:

What is the solution of ?

6(x + 5) = (x + 5)(x + 3)6x + 30 = x2 + 8x + 15

0 = x2 + 2x – 150 = (x + 5)(x – 3)

Problem 5 (Continued)What is the solution of ?

0 = (x + 5)(x – 3)Let’s check x = -5 and x = 3

Undefined!

It checks!

Got it? 5 What is the solution of ?

x = 0

Inverse VariationLesson 11-6

Examples of Inverse VariationThe more workers building a house, the less days it will be unlivable.

The longer the board, the less strength you need to break it.

The more roommates in an apartment, the less rent everyone has to pay.

When one increase, the other decrease (and vice versa).

Inverse Variation in MathKey Concept:“An equation of the form xy = k or y = , where k ≠ 0, is an inverse variation”

The constant “k” is the product of x and y for an ordered pair (x, y)

The more roommates for an apartment, the less everyone will have to pay a month. Say the rate for each person is

$400 and there are 3 people living there. The total rent is $1200.

x = 400, y = 3 and k = 1200

Problem 1Suppose y varies inversely with x, and y = 8 when x = 3. What is an equation for the inverse variation?

xy = k3(8) = k24 = k

The inverse equation would be xy = 24 or y =

Got it? 1Suppose y varies inversely with x, and y = 9 when x = 6. What is an equation for the inverse variation?

k = 54

xy = 54 or y =

Problem 2The weight needed to balance a lever varies inversely with the distance from the fulcrum to the weight. How far away from the fulcrum should be person sit to balance the lever?

The constant “k” is the same for the boy and the elephant. 160x = constant

1000(7) = constant

Problem 2 (continued)The weight needed to balance a lever varies inversely with the distance from the fulcrum to the weight. How far away from the fulcrum should be person sit to balance the lever?

The constant “k” is the same for the boy and the elephant. 160x = constant

1000(7) = constant

160x = 7000x = 43.75

The boy should move 43.75 from the fulcrum to balance the level.

Got it? 20A 120 lb weight is placed on a lever, 5 feet from the fulcrum.

How far from the fulcrum should an 80 lb weight be place to balance the lever?

120(5) = 80x600 = 80x

x = 7.5 ft

Graphs

Problem 3 – Graphing y = Transform this into a table.

X Y

-8 -1

-4 -2

-2 -4

0 Undefined

2 4

4 2

8 1

Got it? Graph y =

Problem 4: Determining Inverse VariationDoes this table show an inverse variation or a direct variation?

Direct will have a constant of .

Inverse will have a constant of xy.

Try both constants and see what one is true.

Direct Variation: ?? Yes, this is direct variation. Inverse Variation: 3(-15) = 4(-20) = 5(-25)?? No, it is not inverse.

X Y

3 -15

4 -20

5 -25

Problem 4: Determining Inverse VariationWhat is the equation for this table?

Direct variation formula is y = kx, where k is the constant.

To find the constant divide y by x.

In this case, k = -5.

The equation is y = -5x

X Y

3 -15

4 -20

5 -25

Got it? 4Does this show inverse or direct variation? Give the equation of the table.

A.

B.

X Y

4 -12

6 -18

8 -24

X Y

4 -12

6 -8

8 -6

Direct; y = -3x

Inverse; xy = -48

Problem 5 – Real LifeDoes this situation show inverse or direct variation?

a. The cost of a $120 boat rental is split among several friends.

The more friends you have, the less each person has to pay.This is inverse variation.

b. You download several movies for $14.99 each.

The more movies you buy, the more it will cost.This is direct variation.

Graphing Rational FunctionsLesson 11-7

Rational FunctionsInverse variations are radical functions.

Ex.On any trip, the time you travel in a car varies inversely with the car’s average speed. The function t = represents the time it takes to travel 60 miles at different rates.

t = is a radical function.What is the excluded value?

r = 0

Problem 1: Finding excluded values

What is the excluded value?

A. f(x) =

x = 2At x = 2, the function is undefined.

B. f(x) = x = -8

At x = -8, the function is undefined.

Comparing Graphsy = y =

The second graph is moved 3 units to the right. If x is 3, then the fraction is undefined. The line x = 3 is an asymptote.

Where is the asymptote?1. y =

x = -9

2. y =

x = -3

3. y = x = 4

Problem 2: Graphing with a Vertical Asymptote

What is the vertical asymptote of the graph y = ? Graph this function.

Got it? 2

What is the vertical asymptote of the graph y = ? Graph this function. **Remember that the fraction is NEGATIVE, so the graph will be the opposite of the last function. **

Asymptote: x = 6

Vertical and Horizontal Asymptotes:

The graph can move right, left, UP and DOWN.

Problem 3: Vertical and Horizontal Asymptotes

What are the asymptotes of the graph f(x) = ? Graph this function.

Vertical asymptote: x = 1Horizontal asymptote: y = -2

Got it? 3What are the asymptotes of the graph f(x) = ? Graph this function.

Vertical asymptote: x = -3Horizontal asymptote: y = -4

In General…If the function is POSITIVE: the graph will be in quadrants 1 and 3.

If the function is NEGATIVE: the graph will be in quadrants 2 and 4.

Vertical asymptote is x = Horizontal asymptote is y =