Rational
Expressions &
Equations/
Reciprocals
Math 20 – Pre-Calculus
Chapter 6
Name:__________________
Class:___________________
1
Algebra and Number
Specific Outcomes
Determine equivalent forms of rational
expressions
(limited to numerators and
denominators that are
monomials, binomials or trinomials).
Perform operations on rational
expressions (limited to
numerators and denominators that are
monomials,
binomials or trinomials).
Solve problems that involve rational
equations
(limited to numerators and
denominators that are
monomials, binomials or trinomials).
General Outcome:
Develop algebraic reasoning and number sense.
Achievement Indicators:
The following set of indicators may be used to determine whether
students have met the corresponding specific outcome
4.1 Compare the strategies for writing equivalent forms of rational expressions to
the strategies
for writing equivalent forms of rational numbers.
4.2 Explain why a given value is non-permissible for a given rational expression.
4.3 Determine the non-permissible values for a rational expression.
4.4 Determine a rational expression that is equivalent to a given rational expression
by
multiplying the numerator and denominator by the same factor (limited to a
monomial or a
binomial), and state the non-permissible values of the equivalent rational
expression.
4.5 Simplify a rational expression.
4.6 Explain why the non-permissible values of a given rational expression and its
simplified form
are the same.
4.7 Identify and correct errors in a simplification of a rational expression, and
explain the
2.1 reasoning.
5.1 Compare the strategies for performing a given operation on rational expressions
to the
strategies for performing the same operation on rational numbers.
5.2 Determine the non-permissible values when performing operations on rational
expressions.
5.3 Determine, in simplified form, the sum or difference of rational expressions with
the same
denominator.
5.4 Determine, in simplified form, the sum or difference of rational expressions in
which the
denominators are not the same and which may or may not contain common factors.
5.5 Determine, in simplified form, the product or quotient of rational expressions.
5.6 Simplify an expression that involves two or more operations on rational
expressions.
(It is intended that the rational equations be those that can be simplified to linear
and
quadratic equations.)
6.1 Determine the non-permissible values for the variable in a rational equation.
6.2 Determine the solution to a rational equation algebraically, and explain the
process used to
solve the equation.
6.3 Explain why a value obtained in solving a rational equation may not be a
solution of the
equation.
6.4 Solve problems by modelling a situation using a rational equation.
2
Big Ideas:
By the end of this unit you should be able to . . .
 FACTOR rational expressions in order to
simplify them
 Determine for which value(s) of x an expression
is undefined (will usually involve factoring)
 Add and subtract rational expressions which
usually involves finding common denominators
and making equivalent expressions
 Multiply and divide rational expressions which
usually involves factoring, simplifying and then
multiplying
 Solve rational EQUATIONS
 Create a system of rational equations from a
word problem and solve
3
6.1 Rational Expressions
Review of Fraction Basics
Fractions and “0”
Simplifying Rational Expressions / Non-Permissible Values
Rational Expression –
Non-permissible values –
Example: For each rational expression, determine all the non-permissible values.
5x
5x
x2  5x  6
a)
b)
c)
4 xy 2
x( x  5)
x 2  36
4
Rational expressions can be simplified by:
Example: Simplify each rational expression. State the value(s) of any restrictions.
5x
5x
12  3x
a)
b)
c)
2
4 xy
x( x  5)
( x  5)( x  4)
e)
x2  5x  6
x 2  36
f)
x 2  81
9 x
g)
Homework
Pg. 317 (#1-4, 6-8)
5
3x 2  8 x  3
3x 2  15 x  18
6.2 Multiplying and Dividing
Rational Expressions
Review of Steps to Multiplying or Dividing Fractions
Simplify each of the following.
(a)
5 3

8 7
(b)
4 2

7 3
(c)
2 2

3 3
(d)
8
3
9
(e)
5 15

6 16
(f)
3 4
 6
5 5
(g)
3 5 8 5
  
4 6 9 12
6
To Multiply Rational Expressions
Example: Multiply and write your solution in simplest form. Identify all non-permissible
values.
a)
3x3 y 3

y2 6x
b)
x 1
x2
 2
x  5x  6 x  5x  4
c)
x 2  16 x  64 x  4

x2  4x
x 8
d)
2 x 2  x  1 4 x 2  28 x  48

x 2  2 x  3 2 x 2  13x  7
7
2
To Divide Rational Expressions
The Non-Permissible Values in Division of Rational Expressions
For example, in:
A
A C B
 
B D C
D
the non-permissible values are B  0, D  0 and C  0
Example: Divide and write your solution in simplest form. Identify all non-permissible
values.
5 x 4 25 x 2

a)
2
6
x 2  3x  18 x 2  3x  2

c) 2
x  6 x  9 x 2  8 x  15
x  2 2x  4

b)
4x  5 4x  5
8
Example: Simplify the following expression and identify any non-permissible values
4 x  20 x  4
x6
 2

x  6 x  25 2 x  8
Homework
pg. 327 (#1,2, 4, 7,8,12)
9
6.3 Adding and Subtracting
Rational Expressions
Review of Steps to Adding or Subtracting Fractions
Simplify. Express all answers in simplest form.
(a)
3 2

7 7
(b)
2 5

3 8
(c)
3 4

7 21
(d)
4 1

9 4
(e)
3 2 5
 
4 3 12
(f)
35 5 1
 
36 18 3
10
To Add or Subtract Rational Expressions
Case 1: Denominators are the same
Case 2: Denominators are different
Example: Determine each sum or difference. Identify all non-permissible values.
a)
2 x  4 7 x  10

x2  9 x2  9
b)
10a  5 3a  2

ab
ab
Example: Identify the lowest common denominator for each group of rational
expressions.
a)
1 1 1
, ,
2 x 3x y
b)
1
1
1
,
,
x  5  x  5 x  4   x  4  x  7 
11
c)
1
1
, 2
x  6 x  9 x  8 x  15
2
Example: Determine each sum or difference. Identify all non-permissible values..
2x 4 3
 
a)
xy x 2 x
y 2  20 y  2
b) 2

y 4 y2
1
1
 2
c)
4  x x  8 x  16
1
x
d)
1
x
x
1
Homework
pg. 336 (#1,3-10)
12
6.4 Rational Equations
Rational Equation –
Steps to Solving a Rational Equation:
Example: Solve each of the following equations. Identify all non-permissible values.
(a)
5 3 9
 
2x 4 4x
(b)
13
x  2 x 1

x3 x2
(c)
5x
3x  8
2
x4
x4
(d)
14
x
x 1
5


x  1 x  4 x 2  3x  4
Example: Colin rows his boat 24 km downstream and back to where he began. When the
average speed of the current is 2 km/h, Colin can complete the journey in 9 hrs.
What is Colin’s average rowing speed in still water?
Downstream
Distance (km)
Average speed (km/h)
Time (h)
15
Upstream
Example: Katie mows the lawn in 40 min. When Will and Katie work together, they can
mow the lawn in 24 min. How long would it take Will to mow the lawn on his
own?
Time
Fraction of lawn
mowed by Katie
Fraction of lawn
moved by Will
1
2
24
Homework
pg. 348 (#1-4, 12-14)
16
Fraction of lawn
mowed working
together
7.4 Reciprocal Functions
INVESTIGATION:
For each given value of f(x), determine the corresponding value
1
of
:
f x
af
f  x
-10 000
-10
-1
-0.0001
0
0.0001
1
10
10000
Observations . . . .
17
1
f  x
IN GENERAL:
af
For any function, y  f x , the graph of the reciprocal function, y 
1
, can be found
f x
af
using the following general rules:

Where the value of the original function is zero, the value of the
reciprocal function is _________________, and a vertical asymptote
exists.

Where the value of the original function is a very small positive number,
the value of the reciprocal function is _____________________________.

Where the value of the original function is a very small negative
number, the value of the reciprocal function is
_____________________________.

Where the value of the original function is a very large positive number,
the value of the reciprocal function is _____________________________.

Where the value of the original function is a very large negative
number, the value of the reciprocal function is
_____________________________.

Where the value of the original function is 1, the value of the reciprocal
function is _______.

Where the value of the original function is –1, the value of the reciprocal
function is _______.

These points, where the original function and reciprocal function have the
same unchanged value, are called
18
Reciprocal function –
Vertical Asymptote –
Horizontal Asymptote –
Invariant Point(s) –
af
Example: The graph of y  f x is shown below. Use this graph to draw the graph of
1
y
.
f x
af
19
af
Example: A function is defined by the expression f x  x  2.
(a) Find the equation of its reciprocal function, y 
1
.
f x
af
af
(b) Sketch the graph of f x  x  2. Then use the graph of y = f(x) to sketch the graph
1
of y 
on the same grid.
f ( x)
(c) For the reciprocal function, y 
i. Domain:
1
, determine:
f x
iii. Vertical Asymptote:
af
ii. Range:
iv. Horizontal Asymptote:
20
Example: A function is defined by the equation y  x 2  4. Use the graph of this
1
function to sketch the graph of y  2
then analyze this reciprocal function by stating
x 4
the domain, range, zeros, asymptotes, and the coordinates of any max/min points.
i.
Domain:
ii.
Range:
iii.
Zeros
iv.
max/mins:
v.
Vertical Asymptote:
vi.
Horizontal Asymptote:
Homework:
pg. 403 (#1-9)
21
22