Download - Rational expressions and equations
![Page 1: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/1.jpg)
Chapter 6
RATIONAL EXPRESSIONS AND
EQUATIONS
![Page 2: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/2.jpg)
Chapter 66.1 – RATIONAL EXPRESSIONS
![Page 3: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/3.jpg)
RATIONAL EXPRESSIONS
A rational expression is an algebraic fraction with a numerator and a denominator that are polynomials.
What is a rational number? What might a rational expression be?
Examples:
![Page 4: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/4.jpg)
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.
![Page 5: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/5.jpg)
EXAMPLE
For each rational expression, find its non-permissible values:a)
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:b) (you will need to factor the
denominator)
![Page 6: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/6.jpg)
SIMPLIFING RATIONAL EXPRESSIONS
Recall:
To simplify rational expressions, we need to find any common factors in the numerator and denominator.
![Page 7: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/7.jpg)
EXAMPLE
Simplify, and state the non-permissible values:
Need to factor the numerator and the denominator:
Try it:
![Page 8: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/8.jpg)
EXAMPLE
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.
a) 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.
b)
Recall: difference of squares
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
![Page 9: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/9.jpg)
Independent Practice
PG. 317-321, #2, 4, 6,
![Page 10: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/10.jpg)
Chapter 66.2 – MULTIPLYING
AND DIVIDING RATIONAL NUMBERS
![Page 11: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/11.jpg)
MULTIPLYING RATIONAL NUMBERS
What is the rule for multiplying fractions?
What are the non-permissible values?
![Page 12: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/12.jpg)
EXAMPLE
Multiply, and simplify the expression. Identify all non-permissible values.
First, you need to factor as much as you can.
Cancel any common factors
What are the non-permissible values?
![Page 13: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/13.jpg)
TRY IT
![Page 14: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/14.jpg)
EXAMPLE
Determine the quotient in simplest form.
What’s the rule for dividing fractions?
Why is –5 a non-permissible value?
![Page 15: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/15.jpg)
TRY IT
![Page 16: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/16.jpg)
EXAMPLE
Simplify:
Why did I factor out the –1 from (3 – 2m)?
![Page 17: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/17.jpg)
TRY IT
![Page 18: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/18.jpg)
Independent Practice
PG. 327-330, #2, 8, 12, 14, 15, 16, 19
![Page 19: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/19.jpg)
Chapter 6
6.3 – ADDING AND SUBTRACTING
RATIONAL EXPRESSIONS
![Page 20: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/20.jpg)
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:
![Page 21: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/21.jpg)
EXAMPLE
What will our common denominator be?
What will our common denominator be?
![Page 22: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/22.jpg)
TRY IT
![Page 23: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/23.jpg)
Independent Practice
PG. 336-340, #3, 5, 6, 7, 10, 14, 15, 25
![Page 24: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/24.jpg)
Chapter 66.4 – RATIONAL EQUATIONS
![Page 25: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/25.jpg)
RIDDLE
Diophantus of 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?
![Page 26: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/26.jpg)
EXAMPLE
Solve for z:
First, factor all of the denominators. What will the common denominator be?
Multiply each side by the common denominator.
What are the non-permissible values?
z ≠ 2, -2 So, z = 5 is an acceptable answer.
![Page 27: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/27.jpg)
TRY IT
Solve for y:
![Page 28: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/28.jpg)
EXAMPLE
Solve the equation. What are some non-permissible values?
The non-permissible values are 2 and -2.
![Page 29: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/29.jpg)
EXAMPLE
Solve the equation. What are some non-permissible values?
Does k = –2 work? Why or why not?
![Page 30: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/30.jpg)
TRY IT
Solve the equation. What are some non-permissible values?
![Page 31: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/31.jpg)
EXAMPLE
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 papersJeff can deliver 1/50th of the papers
So, while working together for x minutes we can say that:
Check!
![Page 32: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/32.jpg)
EXAMPLE
In a dogsled race, the total distance was 140 miles. Conditions were excellent on the way to Flin Flon. 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 Flin Flon?Each half of the race is 70 miles. Lets call the speed that the winner was going on the way to Flin Flon x. 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.
![Page 33: Rational expressions and equations](https://reader036.vdocuments.us/reader036/viewer/2022062520/568164d5550346895dd70ffa/html5/thumbnails/33.jpg)
Independent Practice
PG. 348-351, #1, 3, 5, 9, 13, 14, 19, 22