mat 125 – applied calculus 1.2 review ii. today’s class we will be reviewing the following...
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MAT 125 – Applied Calculus1.2 Review II
1.2 Review II
2 Today’s Class
We will be reviewing the following concepts: Rational Expressions
Other Algebraic Fractions
Rationalizing Algebraic Fractions
Inequalities
Absolute Value
Dr. Erickson
1.2 Review II
3 Rational ExpressionsQuotients of polynomials are called rational expressions.
For example
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3 8
2 3
x
x
2 35 2
4
x y xy
x
2
5ab
1.2 Review II
4 Rational ExpressionsThe properties of real numbers apply to rational expressions.
ExamplesUsing the properties of number we may write
where a, b, and c are any real numbers and b and c are not zero.Similarly, we may write
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1xy x y x x
ty t y t t
( 2)( 5) ( 2)( 2,5)
( 2)( 5) ( 2)
x x xx
x x x
1.2 Review II
5
Example 1
Simplify the expression(s).
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2 2 2 2 2
2 3 2 4
2 3 9 (1 ) (2) (2 )(2)(1 )(2 )
2 3 ( 1)
a ab b x x x x
ab b x
a. b.
1.2 Review II
6 Rules of Multiplication and Division
If P, Q, R, and S are polynomials, then
Multiplication
Division
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( , 0)P R PR
Q SQ S QS
( , , 0)P R P S PS
Q R SQ S Q R QR
1.2 Review II
7
Example 2
Perform the indicated operation and simplify
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22 2 2
2 2 2 3
26 9 3 6 3 4 4.
6 2 7 3
y xx x x x xy y
x x x x x y x y
a b.
1.2 Review II
8 Rules of Addition and Subtraction
If P, Q, R, and S are polynomials, then
Addition
Subtraction
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( 0)P Q P Q
RR R R
( 0)P Q P Q
RR R R
1.2 Review II
9 Addition and Subtraction with unlike Denominators
Find the least common denominator (LCD)
Multiply each term by what is missing from the LCD
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1.2 Review II
10
Example 3
Perform the indicated operation and simplify
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1 2
3 5 1
xxa b xe
ea b x
a. b.
1.2 Review II
11 Other Algebraic Fractions
The techniques used to simplify rational expressions may also be used to simplify algebraic fractions in which the numerator and denominator are not polynomials.
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1.2 Review II
12
Example 4
Simplify
53 2
2
1 12 1
6 2 11 21
xx yx x x
x xxy
a. b.
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1.2 Review II
13 Rationalizing Algebraic Fractions
When the denominator of an algebraic fraction contains sums or differences involving radicals, we may rationalize the denominator.
To do so we make use of the fact that
2 2
a b a b a b
a b
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1.2 Review II
14
Example 5
Rationalize the denominator
2
1 3 2
a a b
a a b
a. b.
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1.2 Review II
15
Example 6Rationalize the numerator
1 3
3
x x x
x
a. b.
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1.2 Review II
16 Properties of Inequalities
If a, b, and c, are any real numbers, then
Property 1 If a < b and b < c, then a < c.
Property 2 If a < b, then a + c < b + c.
Property 3 If a < b and c > 0, then ac < bc.
Property 4 If a < b and c < 0, then ac > bc.
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1.2 Review II
17
Example 7
Find the set of real numbers that satisfy –3 2x – 7 < 9
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1.2 Review II
18
Example 8
Solve the inequality
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3 4 2 2 0.x x
1.2 Review II
19
Example 9
Solve the inequality
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2 34.
1
x
x
1.2 Review II
20 Absolute Value
The absolute value of a number a is denoted | a | and is defined by
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0
0
a aa
a a
if
if
1.2 Review II
21 Absolute Value Properties
If a, b, and c, are any real numbers, then
Property 5 | – a | = | a |
Property 6 | ab | = | a | | b |
Property 7 (b ≠ 0)
Property 8 | a + b | ≤ | a | + | b |
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aa
b b
1.2 Review II
22
Example 10
Evaluate the expressions.
a. | -4 | + 4
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6 e b.
1.2 Review II
23
Example 11
Evaluate the inequalities.
a. | x | 2 b. | 2x – 3 | 8
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1.2 Review II
24 Next Class We will discuss the following concepts:
The Cartesian Coordinate System
The Distance Formula
The Equation of a Circle
Slope of a Line
Equations of Lines
Please read through Section 1.3 – The Cartesian Coordinate System and Section 1.4 – Straight Lines in your text book before next class.
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