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Rational expressions and equations. Chapter 6. 6.1 – rational expressions. Chapter 6. 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:.

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rational expressions
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.


non permissible values
Non-permissible values

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.


For each rational expression, find its non-permissible values:


When you have a denominator that is broken up into factors (numbers or expressions that multiply together), then you need to let each factor be equal to zero to find the non-permissible values:

x = 0 (non-permissible value)

2x – 3 = 0

 2x = 3

 x = 3/2 (non-permissible value)

Try it:


(you will need to factor the denominator)

simplifing rational expressions


To simplify rational expressions, we need to find any common factors in the numerator and denominator.


Simplify, and state the non-permissible values:

Need to factor the numerator and the denominator:

Try it:


Consider the expression .

a) What expression represents the non-permissible values for x?

b) Simplify the rational expression.

c) Evaluate the expression for x = 2.6 and y = 1.2.

Let 8x – 6y = 0

 8x = 6y

 x = 6y/8 = 3y/4

 x ≠ 3y/4

Some examples, then of non-permission values are: (3/4, 1), (3/2, 2), (9/4, 3), and so on.


  • c) Make sure that the values are permissible:
  • 3y/4 = 3(1.2)/4) = 0.9 ≠ 2.6
  • Value is fine.
  • You can use either expression (using the simplified version will be easier):
  • (4(2.6) + 3(1.2))/2 = 7

Recall: difference of squares

pg 317 321 2 4 6
Pg. 317-321, #2, 4, 6,

Independent Practice

multiplying rational numbers
Multiplying rational numbers

What is the rule for multiplying fractions?

What are the non-permissible values?


Multiply, and simplify the expression. Identify all non-permissible values.

Cancel any common factors

First, you need to factor as much as you can.

What are the non-permissible values?


Determine the quotient in simplest form.

What’s the rule for dividing fractions?

Why is –5 a non-permissible value?



Why did I factor out the –1 from (3 – 2m)?

adding and subtracting
Adding and subtracting

What’s the rule for adding and subtracting fractions?

Adding and subtracting rational expressions is the same. If there is a denominators are the same, we need to add the numerators. If the denominators are not equal then we need to find a common denominator.

For instance:


What will our common denominator be?

What will our common denominator be?


Diophantusof Alexandria is often called the father of new algebra. He is best known for his Arithmetica, a work on solving algebraic equations and on the theory of numbers. Diophantus extended numbers to include negatives and was one of the first to describe symbols for exponents. Although it is uncertain when he was born, we can learn his age when he died from the following facts recorded about him

... his boyhood lasted of his life; his beard grew after more; he married after more; his son was born 5 years later; the son lived to half his father’s age and the father died 4 years later.

How many years did Diophantus live?


Solve for z:

First, factor all of the denominators. What will the common denominator be?

Multiply each side by the common denominator.

z ≠ 2, -2

 So, z = 5 is an acceptable answer.

What are the non-permissible values?

try it4
Try it

Solve for y:


Solve the equation. What are some non-permissible values?

The non-permissible values are 2 and -2.


Solve the equation. What are some non-permissible values?

Does k = –2 work? Why or why not?

try it5
Try it

Solve the equation. What are some non-permissible values?


Two friends share a paper route. Sheena can deliver the papers in 40 minutes, Jeff can cover the same route in 50 minutes. How long, to the nearest minute, does the paper route take if they work together?

In one minute:

Sheena can deliver 1/40th of the papers

Jeff can deliver 1/50th of the papers

So, while working together for x minutes we can say that:



In a dogsled race, the total distance was 140 miles. Conditions were excellent on the way to FlinFlon. However, bad weather caused the winner’s average speed to decrease by 6 mph on the return trip. The total time for the trip was 8.5 hours. What was the winning dog team’s average speed on the way to FlinFlon?

Each half of the race is 70 miles. Lets call the speed that the winner was going on the way to FlinFlonx. Then, on the way back, he was travelling at a speed of x – 6.

Recall that the formula for time is t = d/v

On the way there the winner had a time of:

On the way back, he had a time of:

So, the total time (or 8.5 hours) is represented by the formula:

Try solving for x.