**Unit 6: Rational Expressions and Equations**

Application/Extend questions

1.A school art class is planning a day trip to the Glenbow Museum in Calgary. the cost of the bus is $350 and admission is $9 per student.

a)Write a rational expression that could be used to determine the cost per student if n students go on the trip.

b) Use your expression to determine the cost per student if 30 students go.

**Examples of simplifying Rational Expressions**

1. (x+5)

2(x+5)(x-5)

(x+5)(x+5)

2(x+5)(x-5)

X+5 x+5

2(x-5) 2X-10

Solution to Question #2

(16x -1)(6x -x-12)

2(4x-1)92x-3)

(4x+1)(4x-1)(2x-3)(3x+4)

2(4x-1)(2x-3)

(4x+1)(3x+4)

2

12x +16x+3x+4

2

12x +19x+4

2

6x + x+2

Soluion to Question #1

a) 350+9n

n

b) 350+9(30)

30

**A rational expression is an algebraic fraction with a numerator and denominator that are polynomials.**

To simplify rational expressions:

1. Factor

2.Restrictions

3.Simplify

To find restrictions:

First, we have to ask what are non- permissible values?

Values that make the denominator zero.

to determine them, set the denominator equal to zero.

To simply:

Divide the numerator and denominator by factors that are common to both.

To simplify rational expressions:

1. Factor

2.Restrictions

3.Simplify

To find restrictions:

First, we have to ask what are non- permissible values?

Values that make the denominator zero.

to determine them, set the denominator equal to zero.

To simply:

Divide the numerator and denominator by factors that are common to both.

Practice Problems:

1. The volume of a rectangular prism is (2x³ + 5x² -12x) cubic centimetres. If the length of the prism is (2x-3) centimetres and its width is (x+4) centimeters, what is an expression that would describe the height of the prism?

2.

How to divide rational expressions:

ex. 3:

1. Make sure expressions are in completely factored form

2. Once factored, write out your non-permissible values

3. Multiply the equation by its

reciprocal

4. Multiply numerator by numerator

5. Multiply denominator by denominator

6. Cancel/reduce left over like factors

7. Simplify the equation, keeping it in factored form

Solution to Problem #2

2y²-7y-15

÷

4y²-9 x 12

2y²-10y 6

Solution to Problem #1

(2x³ + 5x² -12x) = h

(x+4)(2x-3)

**Multiplying and dividing rational expressions has similar procedures as multiplying and dividing fractions.**

**ex. 1 :**

1. Reduce or cancel any like binomial factors You cannot cancel part of a binomial!

2. Multiply leftover numerator by numerator

3. Multiply leftover denominator by denominator

4. Reduce the equation if possible, keep the equation in factored form!

ex 2:

1. Make sure your expressions are in completely factored form

2. Once you have factored, write out your non-permissible values

3. Cancel/ reduce any binomial like factors

4. Simplify the equation, keep it in factored form!

1. Reduce or cancel any like binomial factors You cannot cancel part of a binomial!

2. Multiply leftover numerator by numerator

3. Multiply leftover denominator by denominator

4. Reduce the equation if possible, keep the equation in factored form!

ex 2:

1. Make sure your expressions are in completely factored form

2. Once you have factored, write out your non-permissible values

3. Cancel/ reduce any binomial like factors

4. Simplify the equation, keep it in factored form!

**How to multiply rational expressions:**

You can reduce any numerator with any denominator AS LONG AS IT'S MULTIPLICATION

Reciprocal- the inverse of a number. To get the reciprocal, divide 1 by the number. If a fraction, switch the numerator with the denominator.

DO NOT SWITCH

THE SIGNS!

ex. 5 -----> 1/5

2/5 ----> 5/2

Look at the first and last steps to determine the non-permissible values.

**Multiplying and Dividing**

Rational Expressions (6.2)

Rational Expressions (6.2)

**By: Raven, Lauren, Danielle, and Natasha**

**Solving Rational Equations (6.4)**

**3(x-y)**

(x-1) (x+5)

(x-1) (x+5)

**(x-5)(x+5)**

15(x-y)

15(x-y)

**3(x-y)**

(x-1) (x+5)

(x-1) (x+5)

**(x-5)(x+5)**

15(x-y)

15(x-y)

**3(x-5)**

15(x-1)

15(x-1)

**x-5**

5(x-1)

5(x-1)

1

5

(2x-3) centimetres

(x+4) centimetres

Volume= (length) (width) (height)

Volume = Height

(Length) (width)

x( 2x² + 5x -12) = 0

x(2x² +8x -3x-12) = 0

x{ 2x( x+4) -3 (x+4)} = 0

x(2x-3) (x+4) = h

(x+4) (2x-3)

x(2x-3)(x+4) = h

(x+4) (2x-3)

x (2x-3) (x+4) = 0

x = h

The height of the prism is x centimetres.

**(x-4)(x+3)**

(x-3)(x+3)

(x-3)(x+3)

**x²-x-12**

x²-9

x²-9

**x²-4x+3**

x²-4x

x²-4x

**(x-3)(x-1)**

x(x-4)

x(x-4)

**(x-4)(x+3)**

(x-3)(x+3)

(x-3)(x+3)

**(x-3)(x-1)**

x(x-4)

x(x-4)

**x-1**

x

x

**x= -3, 0, 3, 4**

3x-2

x3+ 3x² +2x

9x² - 4

3x² + 8x+ 4

1

x

3x-2

x ( x²+ 3x + 2)

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

(3x² + 6x + 2x+ 4)

1

x

3x-2

x (x+1) (x+2)

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

(3x+2)

(x+2)

1

x

3x-2

x(x+1)(x+2)

(3x+2)

(x+2)

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

x

1

1

x+1

x= -2, -2/3, -1, 0, 2/3

**=**

Extend Question 1

Natalie and Samuel can paint the garage in 3hrs together. Alone, Samuel takes 5hrs. How long does it usually take Natalie to paint the garage?

Example 1

FIN

Thank you for watching Rational Expressions

by Lauren, Raven, Danielle and Natasha

Extend Question 2

An airplane flying from Vancouver to Nunavut is flying

against

a wind of 50km/h. Without the wind, the plane would take 1/2hr less. The distance from Vancouver to Nunavut is 1200km. How fast does the plane travel without the wind, to the nearest km/h?

rational equation: an equation containing at least one rational expression

**In order to solve a rational equation, follow these 5 steps:**

Factor each denominator unless already factored

Identify the non-permissible values

Solve by isolating the variable on one side of the equation

Check your answers!

Factor each denominator unless already factored

Identify the non-permissible values

Solve by isolating the variable on one side of the equation

Check your answers!

**Multiply**

**both sides of the equation**

Examples of rational equations:

Remember! Find LCD of both sides of equal sign

LCD=Lowest Common Denominator

**by the LCD**

multiply by the LCD of the denominators: t and 2

cancel the numerators with the denominators

solve equation by factoring: find the product and sum

check each answer (6 and 2)

Remember non-permissible values! In this case, t cannot equal 0.

Example 2

non-permissible values!

remember to cancel the expressions in

both numerator and denominator!

multiply and add everything together, then move to one side of the equals sign

use the quadratic formula if factoring doesn't work!

Check both answers, and keep both if both work!

FOR THOSE WHO WANT IT:

THE CHECK

YAY!

YAY AGAIN!

Example 3

Check each answer

by substitution

x= -5

x= 1

non-permissible values

multiply by LCD after factoring

quadratic equation in the denominator

remember to cancel

move to one side

factor to get answers (if poss. If not,

use quad formula)

REMEMBER! ALWAYS CHECK WITH NON-PERMISSIBLE VALUES, THEN CHECK EACH

ANSWER WITH ORIGINAL EQUATION

Natalie

Samuel

Together

Time to paint (hrs)

x

5

3

Create a table to organise all the info you have:

Fraction painted in 1hr

1/x

1/5

1/3

Fraction painted in x hrs

x/x=1

x/5

x/3

Natalie's fraction painted in x hrs plus Samuel's is equal to both working together's fraction in x hrs.

Hint: Use variables for the things you don't know, like Natalie's time to paint

Natalie's fraction

Samuel's fraction

Both together fraction

CREATE A RATIONAL EQUATION!

Now, simply solve the equation as usual

No non-permissible values since no

x

in denominators!

Form a statement. Natalie takes 7.5 hours alone to paint the garage.

calm

wind

Speed km/h

Time hrs

Distance km

x

x-50

1200/x

1200/x-50

1200

1200

Create an equation, using the time=distance/speed formula

time in no wind

plus half an hour b/c the wind made the other time half an hour longer

time in wind

non-permissible values

remember to cancel

= 0

use solving the square!

Discard negative value b/c it doesn't make sense for the question. Therefore, the plane took 372km/h with no wind.

**-Lauren Diego**

- Natasha Harland

The height of a stack of books is represented by the equation 2y²-7y-15.

2y²-10y

If the number of books is defined by 4y²-9 , what expression can be used to describe the thickness of a dozen books? 6

Express your answer in simplest form.

÷

÷

÷

÷

÷

(2y+3)(y-5)

2y (y-5)

(2y-3)(2y+3)

6

x 12

(2y+3)(y-5)

2y(y-5)

6

(2y-3)(2y+3)

**Adding and Subtracting Rational Expressions (6.3)**

Application Questions

1. A boat is traveling 20 km at a constant rate of x. To save gas, the boat travels the next 15 km at a constant speed reduced by 3 km form the original speed. What is the algebraic expression for the total time of the boat ride?

Simplifying Complex Rational expressions

Steps:

1) Find a common denominator in BOTH the numerator and the denominator of the complex fraction

2) State non permissible values

3) Add or subtract the numerators of both equations in the complex fraction if necessary

4) Write expression with " " sign

5) Complete expression by multiplying the reciprocal

Solution to problem#2

Let x be the amount of words per minute in the first hundred words

Time=distance

speed

Solution to problem #1

Let x be the starting speed of the boat in km/h

Time = distance

speed

**Case 1: Adding and Subtracting rational expressions when denominators are the same**

Steps:

1) Add or subtract the numerators

2) Complete rational expression by writing new numerator over the common denominator in simplest form

Ex.1: Simplify the expression

5t+3 3t+5 5t+3t+5+3 8t+8 4(t+1)

10 10 10 10 5

**Case 2: Adding and subtracting rational expressions when denominators are different**

Steps:

1)Factor the denominator if necessary

2)Identify the non permissible values

3)Multiply the expression to create a common denominator

4)Add or subtract numerators

5)Write in simplest form

1 3 1 3 1 3

x -x-12 x+3 (x+3)(x-4) x+3 (x+3)(x-4) x+3

1 3x-12 3x-11

(x+3)(x-4) (x+3)(x-4) (x+3)(x-4)

2

**x = 4,-3**

x-4

~Danielle Harrington

2

2

Remember to factor

x-4

(x-4)(x+4)

x+4

(x-4)(x+4)

x

(x-4)(x+4)

x-4

(x-4)(x+4)

h

x-4

2x-4

(x-4)(x+4)

(2y+3) (y-5) (6) (12)

2y(y-5) (2y-3) (2y+3)

72

2y(2y-3)

36

y(2y-3)

36

1

÷

x

x 12

=

=

=

=

=

The thickness of one book is 36

y(2y-3)

x cm

(x+4) cm

(2x-3) cm

Apply the order of operations: divide first, multiply last

Factor each expression completely

To divide, multiply by the reciprocal

Cancel any like factors

Reduce the expression if possible

Ex.3: Simplify the complex expression

Ex.2: Simplify the expression

Reduce the expression completely

**}**

Use composition to factor the expression

Cancel any like factors

CHECK

x (2x-3)(x+4) =

2x³+5x²-12x

x( 2x²+8x-3x-12)

x( 2x²+5x-12)

2x³+5x²-12x

x

x-2

You can cross out terms because it is multiplication

x= -4, 4 and 2

You CANNOT cross out terms when preforming addition or subtraction

State the non-permissible values for each question!

x= -4, 3/2

y= 0, 3/2, 5

To check, multiply the length, width, and height to find the volume

1 1

x+4 x-4

x

x -16

1

x+4

x+4

2x

(x-4)(x+4)

x-4

2x

(x-4)(x+4)

2x-4

(x-4)(x+4)

2x-4

(x-4)(x+4)

(x-4)(x+4)

2x

2x-4

(x-4)(x+4)

(x-4)(x+4)

2x

2x

2x-4

2.

First 20 km

20 km

x km/h

20

x

Last 16 km

16 km

x-3 km/h

16

x-3

h

h

16

x-3

20

x

x-3

x

20(x-3)

x(x-3)

16x

x(x-3)

20x-60

x(x-3)

16x

x(x-3)

36x-60

x(x-3)

Hours

An average student can type x words per minute, after typing 100 words the speed decrease by 5 words per minute for the next 200 words. What is the algebraic expression for the total time of typing 300 words?

First 100 words

100 words

x words/min

100

x

Next 200 words

200 words

x-5 words/min

200

x-5

100

x

200

x-5

x-5

x

**Rational Expressions (6.1)**

**-Raven Grenier**

min

min

100(x-5)

x(x-5)

200x

x(x-5)

100x-500

x(x-5)

200x

x(x-5)

300x-500

x(x-5)

min

The average time for typing 300 words is

300x-500

x(x-5)

minutes

The time it takes the boat to travel the full distance is hours.

36x-60

x(x-3)

2. Parallelogram ABFG has an area of (16x

2

-1)

height of (4x-1) units. Parallelogram BCDE has an area of

(6x -6-12) square units and height of (2x-3) units. What is an expression for the are of triangle ABC?Leave your answer in the om ax +bx+c. What are the non-permissible values?

square units and a

2

2

**Note: Equivalent ration expressions are formed by**

multiplying or dividing both the numerator and the denominator by the same value.

multiplying or dividing both the numerator and the denominator by the same value.

2

350+270

30

=

620

30

=

=20.67

It would cost $20.67 per student in 30 students went.

2

2

2

2

2

=

=

=

=

19

2

=

Decomposition:

(6x-x-12)

(6x-9X+8X-12)

3x(2X-3)+4(2X-3)

(2x-3)(3x+4)

=

=

=

2

2

x=1/4 or 3/2

or

x+5=0 x-5=0

x=-5 x=5

x=+-5

2. (x-3)(x+2)

-7x(3-x)

(x-3)(x+2)

-7x(3-x)

(x-3)(x+2)

-7x(-1)(-3+x)

x+2

7x

=

=

=

-7x= 0

-7 -7

x= 0

3-x=0

-x=-3

-1 -1

x=3

3. 15x y (x-8)(x+2)

5x y(2x+2)(x-8)

3x y(x-8)(x+2)

(2x+2)(x-8)

3x y(x+2)

2x+2

3x y+6x y

2x+2

=

=

=

4

2

2

2

2

3

2

x-8=0 2x-2=0

x=8 2x = 2

2 2 x=0 x=1

y=0

c

A

D

E

F

G

B