Download 5.3C Adding and Subtracting Rational Expressions

5.3C Adding and Subtracting Rational Expressions
Objective:
A.APR.7: Understand that rational expressions form a system analogous to the rational
numbers, closed under addition, subtraction, multiplication, and division by a
nonzero rational expression; add, subtract, multiply, and divide rational expressions.
For the Board: You will be able to add and subtract rational expressions and simplify complex
fractions.
Anticipatory Set:
A complex fraction contains one or more fractions in its numerator, its denominator, or both.
x3
1
1
x2
x
x
Examples:
3
x4
4x  5
x
7x
There are two approaches to simplifying them.
1. Think of them as a “fancy” division problem.
Steps to solve:
1. Simplify the numerator to a single fraction.
2. Simplify the denominator to a single fraction.
3. Rewrite them as a division problem.
4. Solve by “flipping” the second fraction and multiplying.
2. Solve them as they stand.
Steps to solve:
1. Find a common denominator to all denominators in the problem.
2. Multiply both the fraction in the numerator and the fraction in the
denominator by this common denominator.
3. Simplify the remaining non-complex fraction.
Examples: Simplify each complex fraction.
x2
x  2   3 
x2
 1 
 
3
3
1
x
1.
x
x

x  2 x x 2  2 x

 
1
3
3
x2
x2
 1 
3
3
x
x
x
2
1  x  2x
x
3
1
2.
1
x 1
x 1

x  x x  x
4x  5 4x  5 4x  5
1
1
x  1  4x  5

x
1
x  1  1  x  1

4 x  5 4 x 2  5 x
x
1
1
x 1
x 1

x  x x  x
4x  5 4x  5 4x  5
1
1
x 1 x
x 1
 x 1  2
4x  5 x 4 x  5 x
1
1
1
3.
x3
x  x  3  x  4
x4
x
7x
7x
x  3  7 x  7 x  21

x  4 x  4
x
Open the book to page 330 and read example 5.
Example: Simplify. Assume that all expressions are defined.
x2
a. x  1
b.
x 3
x5
x  2 (x  1)(x  5)
(x  2)(x 5)
x 1 
1

x  3 (x  1)(x  5) (x  3)(x  1)
x5
1
x  3 7x
x  1  7 x  21
x  4 7x
x4
7x
1
3 x

x 2
x 1
x
3 x 6  x 2 2x

2
x 2  2x  1  6  x
x 1
x  1 2x 2x  2
x
x
1
White Board Activity:
Practice: Simplify. Assume that all expressions are defined.
x 1
20
2
a. x  1
b. x  1
6
x
3x  3
x 1
(x  1)
1
20
(x  1)
3(x  1)
1
60
(x  1)(x  1) (x  1)
(x

1)
1

 1 


 10
x
x
(x  1) x
6
3(x  1)
6
(x  1)
(x  1)
1
3(x  1)
1
3
2x(x  2)
1 1

3(x  2)
1
c. x 2x = 2x 

x  4 (x  4) 2x(x  2) 2x(x  4)
1
x  2 (x  2)
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 332 – 333 prob. 13 – 15, 28 – 30, 43 – 45.
For a Grade:
Text: pgs. 332 – 333 prob. 14, 28.