RATIONAL EXPRESSIONS
AND EQUATIONS
Chapter 4
4.1 – EQUIVALENT
RATIONAL EXPRESSIONS
Chapter 4
RATIONAL EXPRESSIONS
What is a rational number? What might a rational expression be?
A rational expression is an algebraic fraction with a numerator and a
denominator that are polynomials.
Examples:
y -1
y 2 + 2y + 1
2
1
x
m
m +1
x2 - 4
NON-PERMISSIBLE VALUES
1
x
What value can x not have?
For all rational expressions with variables in the denominator, we need to define the
non-permissible values. These are the values for a variable that makes an
expression undefined. In a rational expression, this is a value that results in a
denominator of zero.
x-7
x-3
x-7
x2 - 9
x-7
x 2 - 4x + 3
EXAMPLE
a) Write a rational number that is equivalent to 8
12
2
4x
+ 8x
b) Write a rational expression that is equivalent to
4x
a)
8 4×2
=
12 4 × 3
2
=
3
What are some other
equivalent fractions?
b)
4x 2 + 8x
,x¹0
4x
4x 2 + 8x 4x(x + 2)
=
4x
4x
= x + 2, x ¹ 0
Firstly, what are the nonpermissible values?
Can I possibly factor
either the numerator
or the denominator?
We need to write our
non-permissible
values at the end.
EXAMPLE
State the restrictions for each rational expression.
3
4x
a)
4- x
b)
a) Determine the non-permissible
values:
-15
x 2 - 5x
b) Determine the non-permissible
values:
x2 – 5x = 0
x(x – 5) = 0
4–x=0
x=4
x=0
There is only one non-permissible
value, x = 4.
We need to factor.
x–5=0
x=5
Two non-permissibles: x = 0, and x = 5
4x 3
,x ¹4
4- x
-15
, x ¹ 0, 5
2
x - 5x
FACTORING REVIEW
When it’s possible to factor a rational expression, we need to be able to do so.
Removing a common factor
Factoring trinomials
If there is a variable or a number that
is a common factor in all the terms of
an expression, we can “factor it out.”
x2 + 9x + 18
Example: 18x2 + 36x + 42
What numbers multiply to 18, and
add to 9?
 3 and 6!
We can factor out a 6.

18x2
+ 36x + 42 =
6(3x2
+ 6x + 7)
Example: 4x3 + 6x2
We can factor out a 2 and an x2.
 2x2(2x + 3)
x2 + 9x + 18 = (x + 6)(x + 3)
Example:
x2 + 4x – 21
x2 + 4x – 21 = (x + 7)(x – 3)
EXAMPLE
For each of the following, determine if the rational expressions are equivalent.
9
a)
3x -1
and
-18
2 - 6x
a) What would I need to multiply 9
by to get –18?
9
9 -2
=
×
3x -1 3x -1 -2
-18
=
-6x + 2
-18
=
2 - 6x
The expressions
are equivalent!
b)
2 - 2x
4x
and
x -1
2x
b) Another method is to check using
substitution. So, choose a value that you’d
like to put in for x. What is a good value to
choose?
 Let’s try x = 3.
2 - 2(3)
1
=4(3)
3
3 -1 1
=
2(3) 3
The expressions are
not equal for x = 3, so
they aren’t equivalent.
PG. 223-224, #3, 5, 6,
11, 14, 15, 16
Independent
Practice
4.2 – SIMPLIFYING
RATIONAL EXPRESSIONS
Chapter 4
EXAMPLE
Simplify the following rational expression:
What are both 24 and 18 divisible by?
6
What are the nonpermissible values?
EXAMPLE
Simplify the following rational expression:
Can I factor the numerator?
What are the non-permissible values?
Can 5x be divided out of 15x3?
Don’t forget your nonpermissible values at
the end.
EXAMPLE
Simplify the following rational expression:
Non-permissible values:
3m3 – 4m2 = 0
 m2(3m – 4) = 0
m2 = 0
m=0
3m – 4 = 0
3m = 4
m = 4/3
So, m ≠ 0, 4/3
PG. 229-231, #3, 4, 5,
8, 9, 10, 13.
Independent
Practice
4.3 – MULTIPLYING AND
DIVIDING RATIONAL
EXPRESSIONS
Chapter 4
EXAMPLE
Simplify the following product:
Step 1: The first step in rational expression
problems is always to factor. Where can we
factor here?
Step 3: Multiply the numerators and
the denominators together.
Step 2: Find the non-permissible values.
Look at all of the denominators.
Note: the nonpermissible values
stay the same.
S
EXAMPLE
Simplify each quotient.
a)
Step 1: Factor
Step 2: Find the non-permissibles. In
division problems, use each
denominator and the second numerator.
b)
Step 3: Take the reciprocal of the
second expression, and then multiply.
EXAMPLE
Simplify each quotient.
a)
Step 1: Factor
Step 2: Find the non-permissibles.
Why didn’t I use 6w, w + 6, or 9w2?
b)
Step 3: Reciprocal, and then multiply.
EXAMPLE
Simplify the following expression:
PG. 238-239, #1, 3, 5,
6, 7, 9.
Independent
Practice
4.4 – ADDING AND
SUBTRACTING RATIONAL
EXPRESSIONS
Chapter 4
EXAMPLE
Simplify the following sum:
What are the non-permissible values?
Find a common denominator.
Our common denominator will be 8x2.
What will we need to
multiply 4x by to get 8x2?
EXAMPLE
Simplify the following difference:
Find the non-permissible values:
What is the common denominator for
this expression?
EXAMPLE
Simplify the following expression:
Step 1: Factor wherever possible.
Step 2: Non-permissible values.
Can I factor?
Step 3: Find the common denominator.
Multiply all the different
factors together.
PG. 249-250, #4, 5, 6, 7,
8, 13
Independent
practice
EXAMPLE
A jet flew along a straight path from
Calgary to Vancouver, and back again,
on Monday. It made the same trip on
Friday. On Monday, there was no wind.
On Friday, there was a constant wind
blowing from Vancouver to Calgary at
80 km/h. While travelling in still air, the
jet travels at a constant speed.
Determine which trip took less time.
What is the equation for time, when
you have speed and distance?
How fast is the jet’s airspeed for two different trips on Friday?
EXAMPLE
Recall that when you add
fractions you need to have
CONTINUED
a common denominator.
Total for Friday
Total for Monday
EXAMPLE CONTINUED
To be able to compare these two times, we need to have
either the numerator or the denominator be the same. Can
we easily create a common numerator or denominator?
Which denominator is larger?
 So what does that say about T1?
A larger denominator means that the fraction
is smaller. So that means that T1 is a smaller
number, and was a shorter trip.
P. 249-250, #9, 10, 11.
Independent
practice
4.5 – SOLVING
RATIONAL EQUATIONS
Chapter 4
EXAMPLE
Solve the following equation for x:
You can tell that it is an equation problem
and not an expression problem because of
the equal sign. There are a different set of
rules, so it’s important to differentiate.
Step 4: Multiply each numerator by the
whole LCD.
Step 1: Factor.
Step 5: Simplify and Solve. You should
be able to get rid of the denominators.
Step 2: Non-permissibles.
Step 3: What would the LCD be?
However, 3 is a NPV.
What does this mean?
EXAMPLE
When they work together, Stuart and Lucy can deliver flyers to all the homes in their
neighbourhood in 42 minutes. When Lucy works alone, she can deliver the flyers in 13
minutes less time than Stuart when he works alone. When Stuart works alone, how long
does he take to deliver the flyers?
Let y be the time it takes Stuart alone.
 then how long does it take for Lucy?
 Lucy takes (y – 13)
Always consider the fraction of deliveries
that can be made in 1 minute:
Stuart alone:
Lucy alone:
Together:
Does y = 6
make sense?
EXAMPLE
Rima bought a case of concert T-shirts for $450. She kept two T-shirts for herself and
sold the rest for $560, making a profit of $10 on each T-shirt. How many T-shirts were in
the case?
What is the expression for price per t-shirt?
What is the expression for profit per shirt?
Do both these
answers make sense?
EXAMPLE
Solve the equation. What are some non-permissible values?
4k - 1 k + 1 k 2 - 4k + 24
=
k+2 k-2
k2 - 4
4k - 1 k + 1 k 2 - 4k + 24
=
k+2 k-2
k2 - 4
The non-permissible values are 2 and -2.
4k - 1 k + 1 k 2 - 4k + 24
Þ
=
k + 2 k - 2 (k - 2)(k + 2)
Þ (k - 2)( 4k - 1) - (k + 2)( k + 1) = k 2 - 4k + 24
æ k 2 - 4k + 24 ö
æ 4k - 1 k + 1 ö
Þ (k - 2)(k + 2) ç
= (k - 2)(k + 2) ç
è k + 2 k - 2 ÷ø
è (k - 2)(k + 2) ÷ø
æ k 2 - 4k + 24 ö
æ 4k - 1ö
æ k +1ö
Þ (k - 2)(k + 2) ç
- (k - 2)(k + 2) ç
= (k - 2)(k + 2) ç
è k + 2 ÷ø
è k - 2 ÷ø
è (k - 2)(k + 2) ÷ø
EXAMPLE
Solve the equation. What are some non-permissible values?
4k - 1 k + 1 k 2 - 4k + 24
=
k+2 k-2
k2 - 4
Þ (k - 2)( 4k - 1) - (k + 2)( k + 1) = k 2 - 4k + 24
Þ (4k 2 - 9k + 2) - (k 2 + 3k + 2) = k 2 - 4k + 24
Þ 3k 2 - 12k = k 2 - 4k + 24
Þ 2k - 8k - 24 = 0
2
Þ k 2 - 4k - 12 = 0
Does k = –2 work? Why
or why not?
Þ (k - 6)(k + 2) = 0
Þk=6
Þ k = -2
PG. 258-260, #1, 6, 8,
10, 11, 12, 15
Independent
practice