Math 130B
1
I. Prerequisites
You must have a basic knowledge and proficiency in the arithmetic of rational numbers as well as an
understanding of the different sets of numbers in the real number system. A basic understanding of the
different laws and properties of real numbers is also expected.
N: set of natural ( or counting ) numbers
N = { ______} is also called the set of ______
count the number of students that are enrolled in a class, the number of teachers at a University
W: Set of whole numbers (nonnegative integers)
W = { ______} is also called the set of ______
the number of heads in a toss of five coins, the number of accidents during the year at an intersection, the number of
A’s in math 130A
I: Set of integers
I = { ______} consist of negative integers and ______
the amount of money left in your checking account – balance in account (rounded to the nearest dollar),
the number of points that you improved from test I to test II ,
the difference in the salary of a professional baseball player and a pro football player(rounded to nearest dollar)
Look at a ratio of integers, we create numbers like;
9/13, -12/7, 25/(-32), 12/6, -4/2,...
→ numbers of this form are part of the set of rational numbers (fractional numbers)
Q: set of rational numbers (quotient of integers)
Q = { m/n | where m and n are integers, n is not zero }, fractions
Q/: set of irrational numbers
Q / : real numbers that can not be written as a fraction ( ratio of integers)
R: set of real numbers
set that consists of all rational and irrational numbers
Other special sets:
P: set of prime numbers
P = { ______}
E = set of even whole numbers
E = { ______}
O = set of odd whole numbers
O = { ______}
Real Number Line
We talk about the opposite of a number or the negative of a number (additive inverse)
example:
What is the opposite of - 3/7 ? ______What is the additive inverse of 5 ? ______
What is the negative of - 4 ? ______
We can plot real numbers on a number line (the real number line ) –
real numbers represent the distance from the origin.
Plot: -2, 4, and
Which is larger p or q ?
p q
| |
Basic Properties (laws) of Real numbers:
Commutative law
of addition: a + b = b + a
3 + 7 = 7 + 3, 213 + 317 = 317 + 213
of multiplication:ab = ba
13 ( 20 ) = 20 (13 ) , 21(210 ) = 210( 21 )
Associative law
of addition: (a + b)+c = a + ( b + c )
(13 + 29 ) + 11 = 13 + ( 29 + 11 ) , (187 + 291) + 9 = 187 +( 291 + 9 )
of multiplication: (ab)c = a(bc)
(7 • 4 ) • 5 = 7 • ( 4 • 5 ) (14 • 8 ) • 5 = 14 • ( 8 • 5 )
Distributive law
of multiplication over addition: a ( b + c ) = ab + ac
2 ( 3 + 4 ) = 2 • 3 + 2 • 4 2 ( x + 4 ) = 2 • x + 2 • 4
Other Properties:
1. Properties of 1 :
1 • c = c c • ( 1/c) = 1
2. Properties of zero:
0 • c = 0 0 / c = 0 if c ≠ 0c/0 = undefined
3. Closure:
The sum of any two whole numbers will be a whole number
The product of any two whole numbers will be a whole number
Example:
If b and c are whole numbers → a + b is a whole number ? ______a • b is a whole number ? ______
a – b is a whole number ? ______a ÷b is a whole number ? ______
Basic Arithmetic Operations
1. Addition( addends: a + b )
a) two numbers that have like signs
1) ( - 4 ) + ( - 24 ) = ______2) ( 345 ) + ( + 273 ) = ______
3) ( - 2/7 ) + ( - 3/5)4) 2.3 + ( + 4.01) = ______
b) two numbers that have different signs
1) ( - 12 ) + ( + 8 ) = ______2) ( 120 ) + ( - 22 ) = ______
3) 3 ¼ + ( - 2 ½ ) = ______4) ( - 2.3 ) + ( 0.02 ) = ______
Word Examples:
1) At the beginning of the day (6:00 AM) the temperature is 30 o F. By 5:00 PM the temperature has increased by
44o F. What is the final temperature ? ______
By midnight the temperature has risen (– 8o F ). What is the temperature at midnight ? ______
What is the average change in temperature during the 18 hour period ? ______
2) value of a stock at opening is $ 12 ¼ the stock price changes at an average of $ ¼ per hour (per share)
What is the value six hours after the stock market opened ? ______
Additive Inverse:
4 is the additive inverse of ( -4 ) because 4 + ( -4 ) = 0 → we call zero the additive identity.
What is the additive inverse of - 2 ? ______of - ( - 2 ) ? ______
What is the additive inverse of 1/x ? ______of -c ? ______
We also call zero the additive identity because
x + 0 = 0 + x = x → no change occurs.
2. Subtraction ( minuend and subtrahend: a – b )
Rewrite as an addition problem and then use rules of addition. Order of operation should be done from
left to right. May include fractions and decimals.
a - b = a - ( b ) need to find the opposite of b → - b
a – b = a + ( - b ). We have changed the subtraction problem into an addition problem.
6 - ( + 3 ) = 6 + ( - 3 ) = ______
- 24. 2 - ( - 32 ) = - 24.2 + ( 32 ) = ______
3 2/5 - ( 4 ½ ) =______
Word Examples:
1) During a recent contest a group of five individuals compete to see who loses the most weight during a four
week period.
One individual is selected at random and provides the following four results at the end of each week;
+ 4, - 4 ½ - 6 ¼ , + 1
How much did he gain ( lose ) ? ______
2) During a recent football game a quarterback rushes five times. Here are the gains on each of the rushes;
- 12, , - 3, 16, 8
What is his total gain ? ______
3. Multiplication(repeated addition) and Division(repeated subtraction)
( Multiplication: factors and product) ( Division: divisor, dividend, quotient, remainder )
a) 3 ( 4 ) = 3 4 = ( 3) ( 4 ) = 3 x 4 = ( 4 ) + ( 4 ) + ( 4 )
b) 4 ( - 5 ) =
c) 24 ÷6 → how many times can you subtract six until you get to zero
24 – 6 – 6 – 6 – 6 – 6 = 0 → five times.
Rules:
1) the product (or quotient) of two signed numbers having the same (like ) signs will be positive
32 x 57 = ______-23 • (- 3.1) = ______
-366 ÷(-12) = ______ = ______
2) the product ( or quotient ) of two numbers having unlike (different) signs will be negative
2 ½ • ( - 3 2/5 ) = ______( - 2.1) ( 42.3 ) = ______
4.04 ÷( - 0.8) = ______
Word Examples:
1) A worker can box 13 pairs of shoes in boxes every three minutes. How many shoes can he box in 66 minutes ?
2) A child is expected to drink 48 oz. of juice. There are 24 children invited to a party. How many gallons of
juice should be ordered so that there is enough for all 24 children and up to six additional children – in case
unexpected guests arrive ?
3) A fruit can be cut as precise as it needs to be cut. There are 7 individuals that want a taste. A total of five
whole fruits are available (means five apples, five oranges,...). How much does each person get ?
4) At the end of the party only one third of a cake is left. The host cuts the cake into fourths and gives each piece
out. After you get home you decide to eat one-fifth of your piece. What portion of the original cake did you eat?
Order of Operations:
1) If only addition or subtraction appear in the expression – do the operations left to right.
2) If only multiplication or division appear in the expression – do the operations left to right.
3) If you have a mixture of these four operations, do multiplication and division first ( left-to-right)
and then repeat with the addition and subtraction operations.
Examples:
1) 2 + ( -3 ) • (2) - ( - 2 ) = ______
2) - 12 ÷4 – 4 + 2 • ( - 3 ) = ______
3 ) 2 – 2 ( 3 ) = ______
4) 24 + 6 ÷2 = ______
More on these order of operations later. PEMDAS
To finish this first set of notes.
We learned that 0 was called the additive identity because for any two real numbers c, we can find another
real number - c ( additive inverse or opposite) so that c + (-c) = 0
If x is a real number with x ≠0, then there is another real number 1/x so that
x • ( 1/x) = 1.
We say x and 1/x are multiplicative inverses of each other ( reciprocals) and we call 1 the multiplicative identity.
HW: Chapter 1
page 4: 2, 7, 8, 9, 10,
page 7: 1, 5, 10, 16, 17, 20, 24
page 9: 1, 7, 10, 14, 16, 20,
page 10: 3, 5, 11, 18, 22,
page 14: 4, 6, 7, 10, 12, 16, 17, 20
page 15: 1, 8, 16, 19
page 18: 2, 5, 6, 9, 11
page 19: 1, 5, 10
page 20: 1, 4, 6, 10, 15, 20
page 22: 1, 5, 6, 11, 17,
page 24: 4, 9, 13, 20
page 26: 1, 4, 9, 11, 14, 20, 30
page 28: 3, 8, 15, 24
page 33: 1, 4, 8, 11, 13, 18
page 34: 21, 23, 27, 32, 37
Name ______Math 130B - Quiz
1. Which of these sets is called the set of natural numbers
{ 1, 2, 3, 4, 5, …. }, { 0, 1, 2, 3, … }, { …, -2, -1, 0, 1, 2, … } , or none of these ? →______
2. Fill in the blank
______is the smallest whole number
______is the smallest prime number
the additive inverse of ½ is ______
A real number that is not a rational number ( can not be written as a fraction) is called an ______
3. Which of these best represents the commutative law of addition ? ______
3 + 4 = 4 + 3 ( 3 + 4 ) + 5 = 3 + ( 4 + 5 ) or neither
4. Which of these best represents the associative law of multiplication ? ______
3( 4 5 ) = ( 3 4 ) 5 3 4 = 4 3 or neither
5. Complete the statements by using the properties of 1 and 0.
a) c 1 = ______b) c 0 = ______c) c + 0 = ______
6. Find the result of
( - 41 ) + ( - 27 ) = ______( 4/13 ) + ( + 7 / 13 ) = ______
7. Find the sum of
( - 24 ) + ( 38 ) = ______( + 4.3 ) + ( -0.54 ) = ______
8. True or False.
______a) all whole numbers are natural (counting) numbers
______b) The quotient of any two whole numbers is always a whole number
______c) 1 is the called the multiplicative identity
______d) zero does not have a multiplicative inverse
Additional examples.
Find each of the following sums
ex. 368 + ( 9732 ) = ______b. 9847 + ( - 869 ) = ______
ex. At 7:00 AM the temperature was 82 0 F. During the day the temperature rose by 19 o F. At night (till
7:00 AM ) the temperature dropped 34o F. What was the final temperature ?
ex. The price of a stock was 27 ¾ when the stock market opened. Overnight news had excited the
crowd and allowed the price to increase by 4 6/8. During the afternoon session the stock dropped
8 ¼. What was the price of the stock at the end of the day ?
What is the opposite of - 42 ______and when you add them you get ( a number and its opposite) ______
Subtraction
ex. Find the difference of 9872 and 3578. ______
Find the difference of 9706 and 8986 . ______
ex. 12 - ( + 24 ) = ______ex. ( - 45 ) - ( + 28 ) = ______
Products and Quotients
3 4 = 4 + 4 + 4 = ______2 24 = 24 + 24 = ______
3 ( - 5 ) = ( - 5) + ( - 5 ) + ( - 5 ) = ______
( - 12 ) ( 6 / 7 ) = ______ex. ( - 20 ) ( 7 / 15 ) = ______
Exponents
products like ( 2 ) ( 2) ( 2 ) ( 2) ( 2) occur so frequently that we develop a shorter notation to represent this type of product; exponent notation.
( 2) 5 = 2 2 2 2 2 = ______,
2 is called the ______, 5 is called the ______or the ______
Examples:
43 = 4 • 4 • 4 = ______( -2 )5 = (-2) (-2)(-2)(-2)(-2) = ______
Notice that x4 = (x)4 but -x4≠x4
( - 4 ) 2 = (-4) (-4 ) = ______- ( 3 )4 = ______= ______
- ( - 2 ) 4 = ______= ______- 42 = ______= ______
Examples:
x2 • x4 = ______y3 • ( 2y2 ) = ______
( 2x3) ( - 4x4 ) = ______4 • 2x3 = ______
Basic Rules of exponents:
Let x and y be any real numbers and let m and n be any whole numbers.
If x 0, then x0 = ______
So, 30 = ______(-2)0 = ______but - 2 0 = - ( 2 )o = ______
1. x n xm = ______2. ( xn )m = ______
ex. ( - 4x )0 = ______ex. x4 x6 = ______
( x4 ) ( x5 ) ( x6 ) = ______y4 y 12 = ______
more examples:
( 2x2 ) ( 3x4 ) = ______( 4x3 ) ( - 5 x4 ) = _____
( - 2xy3 ) ( 3x2y5 ) = ______( -2x)(3xy2 )( 4x2y3) = ______
xn-m if n > m
2. xn xm = 1 if n = m
if n < m
Examples:
= ______ = ______ = ______
= ______ = ______ = ______
= ______ = ______
Literal Expressions
expressions in which letters are used to represent numbers
2x, 3 + 2a, 1 - 2a + 3c
factors – the numbers or letters being multiplied →
2x has two factors; 2 and x 3xy2 has three factors; 3, x, and y2
terms – the expressions being added or subtracted
2x + y + 3 has three terms x- x2 has two terms
ex. 3xy has three factors but only 1 term ex. 1 + a + 2b + 3c has four terms
Def. Two terms are said to be similar or like if the literal factors of one term are exactly the same as the
literal factors of the other(this includes the exponents).
examples:
2x + y has ____similar terms 2x + x + 2 has _____similar terms
x + y + 2x – y has _____ similar terms x2 + 2x + y + 5x – y has _____ pairs of similar terms
We can add literal expressions by combining similar terms.
ex. 3 + 2x = ______ex. 3xy - 8xy = ______
ex. 4x2 + 8x2 = ______x + 2y + 3x - 3y = ______
Def. A polynomial is a literal expression having one or more terms that contain only nonnegative whole
number exponents ( on the variables ).
Note:
When a polynomial has
only one term we call it a →______
two terms, we call it a →______
three terms, we call it a →______
ex. 3x, - 4, x2y, 2xyz4 ______
ex. 2x + 1 , 4 – 8x , x2 - 9y2 ______
ex. 1 + x + x2 , x + y + z , x2 – 2xy + y2 ______
Degree of a monomial:
2x ______x9 ______8x12 ______
3xy ______12x8y4 ______210 → ______
Degree of a polynomial
1 + x + x3 ______2 + x + x10y ______210 + y8 - 2x5y4 ______
Note: If a polynomial has only
1 term we call it a ______two terms →______three terms →______
more than three terms →______
Products of polynomials.
1) monomial x monomial: done it before when we looked at rules of exponents
( 2x ) ( 4x2 ) = ______( - 3x2y) ( 2x ) = ______( 4x3y2)(2xy3) = ______
2) monomial x polynomial : distributive law
2 ( x + 2 ) = ______2 ( x + y + 3 ) = ______
-3x ( x - 2y ) = ______3x2y ( x2 – 4x + 2xy ) = ______
3 ) binomial x binomial : author discusses this later in your text
( 2x + y ) ( x + 3y ) = ______( x – 4y )( 2x + 3y ) = ______
4) polynomial x polynomial : author discusses this later in your text
( x + 2y ) ( x – 2y – 3 ) = ______( 2x + y ) ( 2x + y + 4xy ) = ______
Order of operations – Please Excuse My Dear Aunt Sally
and grouping symbols
{ } squiggles – braces [ ] brackets ( ) parentheses
Please Excuse My Dear Aunt Sally :
parentheses (grouping symbols) should be done first , exponents second,
multiplication and division have equal value in terms of order – go from left to right
addition and subtraction should be done last – have equal value – go from left to right
5 x 7 + 3 x 8 = ______32 + 4 x 3 = ______
6 x 4 ÷8 x 3 – 4 = ______12 ( 4 – 6 )2÷6 = ______
2 – 3[ x + 2(4 – x )] = ______32 – 4 • 16 + 24 ÷22 = ______
Evaluating literal Expressions
What is the value of the expression ( evaluate )
2x + 3 if
x= 2 → ______x = 0 →______x = - ¼ →______
We call x the variable ( a variable ) of the expression.
What is the value of x2 – xy + y if x = - 2, y = - 1 ? ______
The area of a rectangle of width x and length x +2 is found by ? ______
Find the actual area of the rectangle above when x = 2 ½ .
If x = - 1, then which is larger - x2 or x2 – 1
The following formula can be used to find the area (A ) of a trapezoid; A = ½ ( h )( b + c ) where h = ht. and b and c are the bases of the trapezoid.
Find the area of the trapezoid below a = 12 inches
h = 2.5
b = 8 inches
The distance formula is given by d = rt where r represents the rate and t represents the time. Find d when r = 22 mph and t = 4 ½ hours
The area of a circle is given by A = πr2. Find the area when π = 22/7 and r = 14 meters.
The volume of a sphere is given by V = . Find the volume of a sphere given by π = 22/7 and r = 4.2 centimeters.
Chapter Test – Do the review and chapter test in text (pages 38 – 42 )
In the future I may add an additional test here – so this page and the next are left blank on purpose.
Before we look at equations – let’s look at a couple of topics that’s not quite relevant here but will be later.
Greatest common factor(GCF) and Least Common Multiple ( LCM )
Def. (GCF) The greatest common factor (GCF) of a given set of numbers is the largest number that will divide evenly into
every member of the given set.
Look at 8 and 20.
Factors of 8 ( also called divisors of 8 ): 1, 2, 4, and 8Factors of 20: 1, 2, 4, 5, 10, and 20
The greatest (largest) factor they have in common → GCF of 8 and 20 →GCF(12, 20) = 4
With small numbers we can take the approach above and find the actual factor.
With larger numbers we may want to write the numbers in prime factorization ( product of prime factors)
8 = 2 • 2 • 2 = 2 3 and 20 = 2 • 2 • 5 = 22 • 5
The base ( or bases) they have in common → 2 and the smallest occurring exponent is 2 →So, 22 is the GCF.
Find the greatest common factor
of 42 and 36. ______
of 80 and 140. ______
Find the GCF of ( 40, 60, 90 ) . ______
Def. LCM ( Least Common Multiple )
Given a set of numbers we can find a smallest number into which every member of the given set will divide into. We
call this number the least common multiple of that set of numbers.
Given 8 and 12
What is the smallest number that both 12 and 8 will divide evenly.
We want to find the smallest multiple that they have in common.
If the numbers are small enough →
Multiples of 8; 8, 16, 24, 32, 40,...Multiples of 12; 12, 24, 36, 48, 60,...
The smallest multiple they have in common is 24. LCM ( 8, 12) = 24.
As before we can also find the LCM by using the prime factorization of the numbers
8 = 23 12 = 22 • 3
All the different bases with the largest occurring exponents → 23, 51 → product = 23 • 31 = 24
Find LCM of
24 and 30 7 and 13
LCM ( 25, 40, 30 ) = ______
Chapter 2
First-Degree Equations and Inequalities in one variable.
When we write two number expressions that are meant to be equal we write them in an equation form.
3 + 7 = 10 7 + 2 = 10 7 + x = 10
These three are considered equations but they are different types of equations.
The first is always true → we call it an identity. The second one is always false →we call it a contradiction.
The third one is true sometimes (when x = 3 ) and other times it is false. We call this one a conditional equation.
Identify each of the following equations:
ex. 2x – 3 = ( 3x – 3 ) – x → ______
ex. 3 = 1 + 2 →______ex ¾ = 12/ 16→ ______
1/ 2 = 3/ 5 ______and 4 = 2x + ( 1 – 2x ) ? ______
Most of the equations we will work with are conditional equations.
The statement
“ solve the equation” or “ find the solution”
means find a value or values of the variable that when replaced in the equation gives you a true statement. This
is called the solution ( or a solution ) of the equation.
An equation is said to be a first degree equation if the polynomial is of degree 1. We are interested in 1st degree equations in one variable.
x + 2 = 3 , x = 3, 2 - 3 (x + 2 ) = 3 are all first degree equation. Why ? ______
While x2 + 2x = 3, x – xy + 2 = 0 are not. Why ? ______
Equivalent: two equations are said to be equivalent if their solutions are identical.
What is the solution of each of the following
2x = 6 x + 2 = 5 x = 3
Are they equivalent ?
If you knew they were equivalent, which form would be the easiest to look in terms of finding the solution ?
This allows us to look an equation that is simpler but yet equivalent to the original equation
We want to end up with an equation of the form
x = c
Example:
the following two equations are known to be equivalent equations. Find the solution of the second equation.
1) 2x = 8
2)
Methods for solving first degree equations.
It is probably obvious what the solution is of the following equationsbut let us develop a systematic way of solving the
equation.
Rule 1:
A quantity may be added or subtracted to both sides of an equation and the resulting equation will be equivalent
to the original equation.
x + 2 = 5 x – 4 =0.2
Rule #2:
You may divide both sides of an equation by any nonzero quantity and still have an equivalent equation.
0.2x = 12 - 3x = 1.5
Rule #3:
You may multiply both sides of an equation by any nonzero quantity and still have an equivalent equation.
A ratio of two numbers; x:y or x/y
A proportion is a statement in which two ratios are equal.
If the ratio of boys to girls is 2:3, then how many girls are there in a class that consists of 12 boys ?
If the ratio of miles to kilometers is 5 to 8, then how many miles are there in 100 kilometers ?
A recipe calls for 2 tablespoons of sugar to every cup of flour. How much flower should be used with 15 tablespoons
of sugar.
Other Examples.
1.
2. Two more than twice a number is 44. Find the number.
3. A certain stock is worth one-half its value at the end of the day from its morning value. If the value at the end of the day
is $24 ½, what is its morning value ?
______
4. On a map – two cities are located 4 inches apart. If every inch represents 12 ½ miles. What is the actual distance
between them.
5. 3x + 5 = x + 7
6. x – 6 = 5x - 14
7. 3(y + 1 ) + 3 = 7( y – 2 )
8.
9.
10.
11. Word Problems on page 61 and 62.
12.
13.
Sometimes you may be interested in solving equation not in terms of an actual value but in terms other “literals”
Find x if
14. If I = Prt, solve for r.
15. If s = h – vt – ½ at2, solve for v.
First-Degree Inequalities.
symbols:
> : greater than, : less than , : less than or equal , : greater than or equal
We can use these symbols to compare two quantities. Normally we read from left to right.
Write in words:
3 > - 4 ______- 8 < - 4 ______
13/5 2.6 ______- 1/3 1/3 ______
If x is a whole number, then - x < x ? True or False ______
What if x was a natural number ? ______
We can write a set of numbers in terms of an inequality
x < 4 means ______on a number line we write
x - 2 mean ______and on a number line we write
the symbol ( or ) are used to imply that endpoint is not included : (2, 5 )
while [ or ] imply that the endpoint is included: [ 2, 5 ]
Graph each of the following on a number line
x ≤ -2.1 x >
Write an inequality to represent each of the following
Solving Linear Inequalities in one variable
We can find numbers that satisfy, make an inequality a true statement, i.e., we can find a solution to an inequality.
The idea is the same as when we looked for solutions to an equation. We are allowed to add and subtract equal quantities from both sides of the inequality.
Basic Examples:
Other Examples:
2x – 3 > 2 + x 3x - 2 ( 1 + x) > 4
The main difference between equations and inequalities is comes when you multiply or divide.
If we multiply(or divide) both sides of the inequality, it is just like working with an equation
2x > 4
What happens when we multiply(or divide) by a negative number
we know that 2 < 3. What happens when you multiply both sides by a negative
→ - 2 < - 3 which is a false statement
unless we change the direction of the inequality
In general, when we multiply both sides of an inequality by a negative number the direction of the inequality symbol must
change.
2 – x > 4
5x > 2 – 3 ( 4x + 1 )
More Inequalities
Find the solution of
x + 23 < 10 3x + 8 2x - 4
3x + 2 > 2 ( x + 5 ) - 4 2x – ( 4 – 3x ) > 8
2x – 8 < 4x 2 – 3 ( x – 2 ) x – 8
2.1 - 3(x + 1.4) > 2
Absolute Value
Let x be a real number. The absolute value of x measures the distance of the real number x from zero.
We write | x |.
x if x ≥ 0
| x | = or
-(x) if x < 0
In other words, | x | will be equal to either x or -(x) depending on the value of x.
Numerically Equal:
3 and -3 are not equal in value but we say they are numerically
Same with - 2/3 and 2/3; same with 2.1 and (-2.1)
ex. Simplify by writing without an absolute value
( your answer and the value inside the absolute value should be numerically equal)
a) | 7 ½ | = ______b) | - 40.23 | = ______
c) | - 2 / 3 | = ______d) - | - 4 | = ______
e) True or False
If x is a whole number, then | x | = x →______
If x is an integer, then | x | = x → ______
| x + 3 | = | x | + | 3 ) → ______
Conclusion: | x | ≥ 0; the result is never negative. Also, x2 ≥ ?