College Algebra : An Overview

7/27/2019 College Algebra : An Overview

1/26

ALGEBRA :)

Prerequisites

MsD


2/26

A. Modeling the Real World

To create a model is to translate real-lifesituations into mathematical statements.


3/26

Examples:

1. The water bill in a certain subdivision iscomputed as follows : a flat rate ofPhP 25.00 and PhP 1.50 for every cubic meterused. The cubic meter used is the difference

between the present and the last monthsmeter readings. Suppose the reading lastmonth was 300 c.m. and the reading thismonth is 350c.m., how much is the water bill

for household A?


4/26

If W represents the water bill,P = the present monthsreading andL = last monthsreading, then

W = 25.00 + 1.50 per cubic meter x (P L).

Hence, the water bill of household A isW = PhP25.00 + PhP1.50/c.m. (350c.m.

300c.m.)= PhP 25.00 + PhP1.50/c.m.(50 c.m.)

= PhP 25.00 + PhP 75.00= PhP 100.00

Thus the water bill for this month is PhP 100.00


5/26

2. The student organization is preparing a

tarpaulin to promote their incoming activity.After much canvassing, they found the cheapestrate for the printing of tarpaulin : PhP15.00 persquare feet. They plan to post a tarpaulin that

measures 5 ft. in length and 3 ft. in width. Howmuch will the group pay for the tarpaulin?


6/26

First, compute for the area of the tarpaulin.

The area of a rectangle, which is obviouslythe shape of the tarpaulin is Area = Lengthx Width.

Area = 5 ft. x 3 ft. = 15 sq. ft.

To compute the cost,Cost =( PhP 15.00/sq.ft) x 15 sq. ft.

= PhP 225.00

Thus, the group will have to payPhP 225.00 for the tarpaulin.


7/26

Other examples :

1. Compute the total amount of tuitionand fees of Student A if he plans to take15 units, assuming that the tuition feeis 1000 pesos per unit and the

miscellaneous fee is 5250 pesos. Howmany units can he enroll in if he only hasPhP 15,000 ?


8/26

2. What is the grade point average of Student A

if he got the following grades last semester ?Algebra (3 units) = 3.50P.E. (2 units) = 3.75Laboratory (1 unit) = 2.0Physics lecture (3 unit) = 2.25

3. Suppose Student B got the followingscores : Long quiz 1 = 52, Long quiz 2 =

59, Problem set = 60. What should she getin the prelim exam in order to pass the classduring the prelim period?


9/26

B. Real Numbers

The set of real numbers is comprisedof the set of rational (Q) and irrationalnumbers (Q). A rational number is

a number that can be expressed as aratio a/b where a and b are integers.Repeating but nonterminating numberssuch as .3333 (or 1/3) and

nonrepeating and terminating numberssuch as .75 (or ) are examples ofthese.


10/26

Nonterminating and nonrepeatingdecimals such as 3.14159. . . (pie)cannot be expressed as a ratio and

thus are called as irrational numbers.


11/26

Under the set of rational numbers are integersand fractions (nonintegers). Belonging to theset of integers are negative numbers and wholenumbers. Zero (0) and the counting numbers

(1, 2, ) are called whole numbers.


12/26

1. Properties of Real Numbers

Let a, b and c be real numbers. Then thefollowing are true:

a. Commutative propertya + b = b + a ex : 2 + 8 = 8 + 2

ab = ba ex : 2 x 8 = 8 x 2

Note : from hereon, a b may be written as abor a(b)

b. Distributive propertya(b + c) = ab + ac

ex : 4(3 + 9) = 4(3) + 4(9)


13/26

c. Associative property(a+b) + c = a + (b + c)

ex : (5+3) + 34 = 5 + (3 + 34)(ab)c = a(bc)

ex : (3x6)5 = 3(6 x5)

d. Identity element of Addition : a + 0 = aex : 45 + 0 = 45

Identity element of Multiplication : a(1) =aex : 3(1) = 3

e. Existence of Additive Inverse: a+ (-a) = 0ex : 23 + (-23) = 0

Existence of Multiplicative Inverse : a (1/a)=1ex : 4(1/4) = 1


14/26

f. Closurea + b is also a real numberab is also a real number

ex : 4(5) = 20, which is a real number

also

? Is subtraction under the set of Realnumbers commutative?? Do we have closure property under the setof rational numbers ? Set of irrationalnumbers?


15/26

Real Number Line

Graphing Intervals

Real numbers may be represented in anumber line. The numbers to the left of 0 arethe negative numbers while those in the right arethe positive numbers. Given two distinct numbersa and b, if a is to the left of b then a is less thanb. If a is to the right of b, then a is greater thanb.


16/26

Getting the intersection and union

a. [-2, 7.5) (-, 5]

b. [-2, 7.5) (-, 5]


17/26

Absolute value and distanceLet a be a real number. The

absolute value of a, denoted by |a|, is thedistance of a from 0.

Examples :|-3.5| = 3.5 |10| = 10 |0| = 0

0aifa

0aif0

0aifa

|a|


18/26

a. Properties of absolute value

Note that the absolute value of aproduct is equal to the product of theirabsolute values. Example : |-35| = |-7| x|5| = 35

Also, the absolute value of a quotient isthe quotient of their absolute values. Forexample ,

8

5

8

5

8

5


19/26

b. Distance between points on the real line

Let a and b be two real numbers.The distance between a and b, denoted byd(a,b) is |a - b|. For example : The distancebetween 7 and 21 is |7 - 21| = |-14|= 14.

Exercises : pp. 19-20 : #s 1, 11, 13, 21, 23,25,33, 41, 55, 57, 59, 69, 75, 77


20/26

C. Integer exponents

Let a and n be real numbers. By an

we meanan= aa a (n times)

We call a as the base and n as the exponent.

1. Zero and negative exponents. Note that

a0= 1 and a-n= .

Example : 5-2=

na

1

25

1=

25

1


21/26

2. Laws of exponents. Let a, m and n be real

numbers.am an= am+n ex : x2 x5= x7am an= am- n ex : x2 x5= x-3= 1/x3(am)n= amn ex : (x2)5= x10

(ab)n= anbn ex : (xy)7= x7y7(a/b)n= an/bn ex : (x/y)4= x4/y4

Exercises : pp. 27-29 : #s 5, 7, 21, 29, 33,41, 45, 49, 81, 83


22/26

D. Rational exponents and

radicals 1. Definition of nthroot

Let n be a positive integer. The nthroot of a isequal to b, that is, if bn= a.

In , the number n is the index and a is the

radicand. The symbol is called the radicalsign.

Example : because (x7)3= x21

b=na

n

a

73 21 x=x


23/26

2. Properties of nthroot

ex :

ex :

ex :

if n is odd ex :

= |x| if n is even ex :

nnn yx=xy 23 63 xy=yx

n

n

n

y

x=

y

x 4

2

312

6

y

x2=

y

x8

mnmn x=x 2=64=64 63

x=x

n n 23 3)2(

24 4)2(-

-423


24/26

Note that radicals may be added/subtracted only if

they have the same radicands and indeces. Toadd/subtract them, simply add/subtract theircoefficients.

- = -

= -

=

4

324

4234 4 224

423

42)2(4 423 425


25/26

Definition of rational exponentsBy (a)m/n we mean (a1/n)mor (am)1/nEx 82/3= (81/3)2= 22= 4

(16x6y8)3/2= ((16x6y8)1/2)3= (4x3y4)3= 64x9y12

Using this, we can multiply and simplify thefollowing expression involving rational exponents :

or

2/1x3/1)2/3x(3/1)2

11x(3/1))2/1x)(x((3 xx

x


26/26

Rationalizing the denominatorTo rationalize an expression means to remove

the radical sign in the denominator. To do this,suppose that the exponent of the variable x in thedenominator is m/n, where m/n is a proper fraction.Then multiply the numerator and the denominator

by x(n-m)/n

. Example, torationalize

multiply by x1/33/23 2 x

5=

x

5

xx5=xx5=x

xx5

33/1

3/1

3/1

3/2

Date post:	13-Apr-2018
Category:	Documents
Upload:	chikoy-anonuevo
View:	217 times
Download:	0 times

College Algebra : An Overview

Documents