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Unit 6: Rational Expressions and Equations

By: Raven, Lauren, Danielle, and Natasha
by

Lauren Diego

on 23 May 2014

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Transcript of Unit 6: Rational Expressions and Equations

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.


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!

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)

By: Raven, Lauren, Danielle, and Natasha
Solving Rational Equations (6.4)
3(x-y)
(x-1) (x+5)

(x-5)(x+5)
15(x-y)

3(x-y)
(x-1) (x+5)

(x-5)(x+5)
15(x-y)

3(x-5)
15(x-1)

x-5
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)

x²-x-12
x²-9

x²-4x+3
x²-4x

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

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

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

x-1
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!

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.

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
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