6.1 – Rational Expressions Date:
Key Ideas:
A rational expression is…
Evaluate
0
3 how zero affects division nonpermissible values
When zero is divided by any non-zero real number, …
Evaluate
7
0
Division by zero is undefined because…
3
For the expression 𝑥−2
, what value for x is non-permissible?
What is a non-permissible value?
Write a rule that explains how to determine non-permissible values:

Examples - Determine the non-permissible values for each rational expression a)
4𝑎
3𝑏𝑐
b) 𝑥−1
(𝑥+2)(𝑥−3)
c) 𝑦
2𝑦
2
2
−4 simplifying rational expressions
Rational Expressions can be simplified by: a) Factoring the numerator and denominator if possible b) Dividing the numerator and denominator by all common factors
Examples - Simplify the rational expressions. Keep a running list of nonpermissible values. a)
2𝑥 2
3𝑥−6
+𝑥−10 b)
2𝑦
2 𝑦
+𝑦−10
2 +3𝑦−10
c)
6−2𝑚 𝑚 2 −9
Example - Rational Expressions with 2 variables
For
25𝑥
2
−4𝑦
2
10𝑥−4𝑦 a) a) Write and solve an equation that represents all of the non-permissible values of x. b) Simplify the expression. c) Evaluate for x = 0.8 and y = 1.5 c) b)

6.2 – Multiplying & Dividing Rational Expressions Date:
Key Ideas: multiplication
& division review
Warmup – Simplify a)
3
(
−4
) (
1
2
) b) (
5
8
) (
−4
) c)
15
2
3
÷
2
3
4 d)
5
−1
10
Explain how to multiply fractions:
Explain how to divide fractions:
Examples - Simplify and keep a running list of non-permissible values a) ( 𝑥+3
) (
2 𝑥+1
) b) (
4
4𝑥
2
3𝑥𝑦
) ( 𝑦
8𝑥
2
) c) ( 𝑑
2𝜋𝑟
) (
2𝜋𝑟ℎ 𝑑−2
)

Examples - Simplify and keep a running list of non-permissible values a) 𝑦
2
−9 𝑟
3
−𝑟
× 𝑟
2
−𝑟 b) ( 𝑦+3 𝑥
2
−𝑥−12 𝑥
2
−9
) ( 𝑥
2
−4𝑥+3 𝑥
2
−4𝑥
)
Examples - Simplify and keep a running list of non-permissible values a) 𝑚
2
−6𝑚−7 𝑚
2
−49
÷ 𝑚
2 𝑚
+8𝑚+7
2
+7𝑚
b)
3𝑥+12
3𝑥
2
−5𝑥−12
÷
12
3𝑥+4
×
2𝑥−6 𝑥+4

6.3 – Adding & Subtracting Rational Expressions Date:
Key Ideas: adding & subtracting review
Warmup – Simplify a)
5
6
−
3
8
b) −
2
3
+
4
5 c)
7𝑥+1 𝑥
+
5𝑥−2 𝑥
Write the steps to adding/subtracting fractions:
Examples - Simplify and identify all non-permissible values a)
10𝑦−1
4𝑦−3
−
8−2𝑦
4𝑦−3 b)
2𝑥 𝑥𝑦
+
4 𝑥
2
− 3

Examples - Simplify and identify all non-permissible values
4 a) 𝑝
2
−1
+
3 𝑝+1 b) 𝑥 𝑥−2
2
+𝑥−6
− 𝑥
2 𝑥
2
+6𝑥+5
+4𝑥+3
Example - Simplify and identify all non-permissible values
2−
4 𝑦
4 𝑦− 𝑦
Get a common denominator in both the numerator and the denominator of the complex fraction.

6.4 – Rational Equations Date:
Key Ideas:
A rational equation is an equation containing at least one rational expression.
Remember, when working with an equation, whatever you do to one side, you do to the other side.
Steps to solving rational equations:
1) Factor each denominator if possible.
2) Identify any non-permissible values (and do this throughout).
3) Multiply both sides of the equation by the lowest common denominator in order to eliminate the fractions.
4) Solve the equation.
5) Check your solutions.
Example – Solve 𝑥
4
+
6 𝑥
= 3.5
9
Example – Solve 𝑦−3
−
4 𝑦−6
= 𝑦 2
18
−9𝑦+18

3𝑥
Example – Solve 𝑥+2
5
− 𝑥−3
= 𝑥 2
−25
−𝑥−6 extraneous solutions word problems
When a solution is the same as a non-permissible value, the solution is not valid and is called ‘extraneous’.
Example – Stella takes 4 hours to paint a room. It takes Jose 3 hours to paint the same area. How long will the paint job take if they work together?
Stella
Jose
Together
Time to Paint (hours) Fraction of Work
Done in 1 hour
Fraction of Work
Done in x hours

Example – A train has a scheduled run of 160km between two cities on Vancouver
Island. If the average speed is decreased by 16km/h, the run will take
1
2
h longer.
What is the average speed of the train?
What is the equation triangle for rate, distance, and time?
Distance (km) Rate (km/h) Time (h)
At faster speed
At slower speed