Download SIMPLIFICATION OF RATIONAL EXPRESSIONS

ENGALG1: LECTURE # 5.1
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EQUATIONS
a b
Denotes that the symbols “a” and “b” represent the same quantity or
stand for the same number
This equivalence relation is more commonly known as an EQUATION where “a” and
“b” are members of an equation.
Solving an Equation – process of finding the solution set for the equation
EQUIVALENT EQUATIONS
- set of different equations having identical solution sets
- obtained by any of the following operations:
o adding the same quantity to both sides of the equation
o multiplying both sides of an equation by a non-zero constant
Linear Equations
An equation that contains first degree terms or first degree equation
{x | x _____}
Solution set of equation is obtained by replacing this same
equation by a succession of equivalent equations, for which
the value of the variable is obtained.
A term on one side of an equation may be transferred to the other side as long as its
sign is changed during the process that is called TRANSPOSITION.
General Rule:
All terms with the variable or “unknown” are placed at the left side and all the
other constants at the other side of the equation
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Examples:
1.
2.
3x  2(5  x)  3(2  3x)  5
2x  7
3( x  1) x
 2x 1 

3
4
2
Fractional Equations
Fractions containing the variable or the unknown in their denominators
Example:
1.
2
3
10


x  2 x  1 ( x  2)( x  1)
2.
20
1
2


x 2  25 x  5 x  5
Note: Extraneous Solution or Extraneous Root
- Solution which is not an element of its solution set
Literal Equations (from the word letter)
An equation in which constants are also represented by letters
Examples:
1.
PV  nRT ,
2.
V f  Vo  at ,
3.
4.
d  Vo t 
n?
5.
an  a1  (n  1)d , n  ?
6.
C
5
( F  32),
9
7.
r
IR  nE
, n?
In
a?
1 2
gt , Vo  ?
2
an  a1  (n  1)d , n  ?
F ?
ENGALG1: LECTURE # 5.1
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Translate the following worded statements into mathematical expressions:
1.
The sum of two numbers x and y
2.
8 more than a number k
3.
9 less a girl’s age
4.
9 less than a girl’s age
5.
Half of a boy’s age diminished by 2
6.
Five more than twice the difference of x and y, x being the larger number
7.
The value when the numerator is increased by 8 and the denominator is
decreased by 3 becomes 8/11
8.
The sum of two consecutive integers is 8
9.
The sum of three consecutive even integers is 10
10. The sum of three consecutive odd integers is 10
APPLICATIONS
Word Problems Leading to First Degree Equations
1. Represent the unknown quantity by a symbol, usually the letter “x”
2. Using the conditions presented in the problem, represent a certain quantity by
two different algebraic expressions. At least, one of these expressions must
involve the unknown. Translate sentences into algebraic expressions.
3. Set up an equation by equating the two expressions and then solve for the
variable.
4. Express the answers as required.
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A. Geometric Problems
Example: The length of a rectangle is 13 inches greater than its width. Its
perimeter is 8 feet. Find its dimensions.
B. Mixture Problems
Example: How many ounces of pure nickel must be added to 150 ounces of an allo
containing 70% nickel to make an allow which is 85% pure?
C. Uniform Motion
Example: A man walks to a neighboring town B from town A at a rate of 5 mph
and returns at a slower rate of 4 mph. if it took him 3 hours and 9 minutes for the
entire trip, how far is town B from town A?
D. Age Problem
Example: John is three times as old as his nephew Pete. Four years ago, he was
four times as old as Pete was at that time. How old is Pete?
E. Number Related Problems
Example: One number is five less than another. If their sum is 135, what are the
numbers?
Solve the following Equations for the Variable “x”:
1.
2x  7 x  1 5x  3


3
4
6
2.
3x 2  2( x  3)  4 x  (3x  1)( x  1)
3.
a 2 x 2  3(3  2ax)  (ax  6)2
4.
5( x  3)( x  6)  2( x  2)( x  5)  x 2  2( x  4)2  3(28  x)
5.
(4 x  3)( x  1)
x 3 ( x  1) 2( x  2)2
1  

16
8 2
4
8
ENGALG1: LECTURE # 5.1
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Solve for the specified variable:
1.
e
F

; F ?
mp H  E
2.
a b c
  ; b?
h L n
3.
F  2e
2F

; F ?
m
He  E
4.
5.
2v  T wv2 
H

T ?


25  2
g 
T  P 2a 
b
   2  a  
R  4 v 
a
P?
Solve the following word problems:
1.
Find three consecutive odd integers whose sum is 39.
2.
Find three consecutive numbers of which 4/7 of the sum of the first and the
second number equals the third number decreased by one.
3.
The tens digit of a two-digit number is 5 more than the units digit. If the
numerical value of the number is 8 times the sum of its digits, find the
number.
4.
The length of a rectangle is three times its width. If the length is decreased by
20 feet and the width is increased by 10 feet. The area is increased by 200
square feet. Find the dimensions of the rectangle.
5.
How many gallons of a 55% solution of glycerin in water must be added to 15
gallons of a 20% solution to produce a 40% solution?
6.
A flask contains 500 mL of an 80% alcohol solution in water. How much of
this solution must be drawn off and replaced by water to obtain a solution
having a concentration of 25% alcohol?
7.
Two cars started at the same time to two points 220 miles apart. They travel
toward each other and passed each other at the end of 2 hours and 45
minutes. If the speed of one is 10 mph faster than the other, find the speed
of each.
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8.
Two automobiles travel at the rate of 50 mph and 30 mph respectively. The
faster travels 2 hours more than the other and goes 140 miles farther. Find
the distance travelled by each.
9.
A car left a parking lot at 1:00 pm travelling on a straight road at a speed of
30 mph. At 1:20 pm, another car left the same parking lot and followed the
same route as the one taken by the first car. If the second car overtakes the
first car in 40 minutes, how fast did the second car travel?
10. A man swims downstream from the shore to a certain point at an effective
rate of 10 ft/min. He swims back from this point to the shore at an effective
rate of 4 ft/min. If it took him 14 min altogether, how far out from the shore
did he go?
11. Joan is 3 times as old as her sister. In 3 years, she will be two years more than
twice the age of her sister will be then. What are their ages?
12. Bill is 3 times as old as his kid brother Jack. Four years ago, he was 4 years
less than 5 times Jack’s age at that time. How old are they?