Counting
Numbers/Natural Numbers: numbers we use
for counting, i.e., 1,2,3,4,5,...
Elements of a
set (members): objects or numbers on a
set, i.e., A = {1,2,3,...}
Finite set: set that has a fixed number of elements,
i.e., {1,2,3}
Infinite
set: set without a fixed number of
elements such as natural numbers, i.e., N = {1,2,3,...}
Set-Builder
Notation: uses a variable to represent
the numbers in a set.
Variable: a letter that is used to stand for some
numbers, i.e., B = {1,2,3,...,49} is written in set-builder notation, B ={ x|x
is a natural number less than 50}
Two sets are equal
if they contain exactly the same numbers.
Union of
sets: If A and B are sets, the union
of A and B, denoted AUB, is the set of all elements that are either in A, in B,
or in both. In symbols, AUB = {x|x ЄA or
x ЄB}
Venn
diagrams: diagrams used to illustrate
sets.
Intersection of
sets: If A and B are sets, the
intersection of A and B, denoted A n B, is the set of all elements that are in
both A and B. In symbols, AnB = {x|x ЄA
and x ЄB}
Subsets: if every member of set A is also a member of
set B, then we write AcB and say that A is a subset of B. I.E., {2,3} c {2,3,4}
Empty set: a set with no members and denoted by the
symbol o with a slash through it
Set
symbols: e is a member of
c is a subset of
= equal
U union
Ф it empty set
e with slash - is not a member of
c with slash - is
not a subset of
≠ is not equal to
n intersection
Whole
numbers: the set of natural numbers
together with the number 0 (W)
Integers: the whole numbers together with the negatives
of the counting numbers (Z)
The natural
numbers N = {1,2,3,....}
The whole
numbers W = {0,1,2,3,....}
The
integers Z =
{....-3,-2,-1,0,1,2,3,....}
Rational
numbers: numbers that are written as
ratios or as quotients of integers (Q)
Q = {a/b|a and b
are integers, with b = Ф}
Ex. of rational
numbers: 7, 9/4, -17/10, 0, 0/4, 3/1,
-47/3, and -2/-6
Terminating
decimal: 26/100 = 0.26
4/1 = 4.0
1/4 = 0.25
The single digit
repeats: 2/3 = 0.6666....
The pair of
digits 25 repeats: 25/99 = 0.252525....
The pair of
digits 19 repeats: 4177/990 =
4.2191919....
Rational numbers
are decimal numbers whose digits either repeat or terminate.
Coordinates: numbers corresponding to the points on the
line
Unit: the distance between two consecutive integers
and any tow consecutive integers.
Origin: the point with coordinate 0
Irrational
numbers: numbers that cannot be
expressed as a ration of integers
Real
numbers: the set of numbers that cannot
be expressed as a ratio of integers
Interval of real
numbers: the set of real numbers that
lie between two real numbers, which are called end points of the interval
Interval
notation: used to represent intervals
Absolute
value: a number's distance from 0 on the
number line
Opposites: two numbers that are located on opposite
sides of zero and have the same absolute value.
Opposite of an
opposite: for any number a, - (-a) = a
Absolute value:
|a| = {a i a is positive or -a if a is negative}
Net worth: the total of your debts and assets
Sum of two
numbers with like signs: to find the sum
of two numbers with the same sign, add their absolute values. The sum has the same sign as the original
numbers.
Additive
inverse: the number a and its opposite,
-a have a sum of zero for any a.
Additive inverse
property: for any real number a, there
is a unique -a such that a + (-a) = -a + a = 0.
Sum of two
numbers with unlike signs (and different absolute values): to find the sum of two numbers with unlike
signs, subtract their absolute values.
The sum is positive if the number with the larger absolute value is
positive. The sum is negative if the
number with the larger absolute value is negative.
Subtraction of
real numbers: for any real numbers, a
and b,a - b = a + (-b).
Product: the result of multiplying two numbers
Factors: the numbers multiplied
Products of
signed numbers: to find the product of
two nonzero numbers, multiply their absolute values. The product is positive if the numbers have
the same sign. The product is
negative if the numbers have unlike signs
Multiplicative
Inverse (reciprocal): just as every real
number has an additive inverse or
opposite, every nonzero real number has one too
Multiplicative
inverse/reciprocal: every non-zero has
an opposite 1/a
Multiplicative
inverse property: for any nonzero real
number a, there is a unique number 1/a such that a x 1/a = 1/a x a = 1
Division of real
numbers: for any real numbers a and b
with b not equal to 0, a/b = a x 1/b.
If a/b = c, a is
called the dividend, b is the divisor and c is the quotient.
Division by
zero: a/b is defined only for b is not
equal to 0, such quotients, 5/0, 0/0, 7/0 and 0/0 are said to be defined.
arithmetic
expression: the result of writing
numbers in a meaningful combination with
ordinary operations of arithmetic.
Sum, difference,
product, quotient: an expression that
involves more than operation if the last operation to be performed is addition,
subtraction, multiplication, or division, respectively.
Grouping symbols
(parentheses): indicates which
operations are performed first.
Exponential
expression: for any natural number n and
real numer a, a to the n power = a x a x a x . . . x a.
We call a the base,
n the exponent, and a to the n power an exponential.
Radical
symbol: indicates the non-negative or
principal square root, i.e., 9 is the square root of 9 is 3
Square
roots: If a to the 2nd power = b, then
a is called a square root of b. If a is
greater than or = to 0, then a is called the principal square root of b and we
write the square root of b = a
Order of
operations: expressions in which some or
all grouping symbols are omitted, are evaluated consistently by using a rule.
Order of
Operations
Evaluate inside
any grouping symbols first. Where
grouping symbols are missing use this order.
1.
Evaluate each exponential expression (in order from left to right).
2.
Perform multiplication and division (in order from left to right).
3.
Perform addition and subtraction (in order from left to right).
Algebraic
expression: the result of combining
numbers and variables with the ordinary operations of arithmetic (in some
meaningful way)
Subscript: a symbol such as y, is treated as any other
variable, read as "y one" or "y sub one"
Communtative
property of addition: for any real
numbers a and b, a + b = b + a
Communtative
property of multiplication: for any real
numbers, a and b, ab = ba
Associative
property of addition: for any real
numbers a, b, and c, (a + b) + c = a + (b + c)
Associative
property of multiplication: for any real
numbers a, b, and c, (ab)c = a(bc)
Distributive
property: for any real numbers a, b, and
c, a (b + c) = ab + ac
Additive
identity: addition of 0 to a number does
not change the number
Multiplicative
identity: multiplication of a number to
1 does not change the number
Additive
identity property: for any real number
a, a + 0 = 0 + a = a
Multiplicative
identity property: for any real number
a, a x 1 = 1 x a = a
Additive inverse
property: for any real number a, there
is a unique number -a such that a + (- a) = - a + a = 0
Multiplicative
inverse property: for any nonzero real
number a, there is a unique number 1/a such that a x 1/a = 1/a x a = 1
Multiplicative
property of zero: fpr amu real number a,
0 x a = a x 0 = 0
Term: a single number or the product of a number
and one or more variables raised to powers
Coefficient: the number preceding the variables in a term
Like terms: if two terms contain the same variables with
the same powers, i.e., 3x^2 and 5x^2 are like terms, 3x^2 and 2x^3 are not like
terms
Simplify an
expression: to write an equivalent
expression that looks simpler, but simplify is not a precisely defined term
Equation: a sentence that expresses the equality of two
algebraic expressions, i.e., 2x + 1 = 7 because 2(3) + 1 = 7 is true, 3
satisfies the equation
Solution/root: any number that satisifies an equation
Solution
set: the set of all solutions to an
equation
Solve an
equation: to find its solution set
Properties of
equality: the most basic method for
solving equations
Properties of
equalities
Addition
Property of Equality: adding the same number to both sides of an
equation does not change the solution set to the equation. In symbols, if a = b, then a + c = b + c.
Multiplication
Property of Equality: multiplying both
sides of an equation by the same nonzero number does not change the solution
set. In symbols, if a = b and c is not
equal to 0, then ca = cb.
Equivalent
equations: equations that have the same
solution set
Least common
denominator (LCD): the smallest number
that is evenly divisible by all the denominators
Identity,
Conditional Equation, and Inconsistent Solutions: An identity is an equation that is
satisfied by every number for which both sides are defined. A conditional equation is an equation
that is satisfied by at least one number but is not an identity. An inconsistent equation is an
equation whose solution set is the empty set.
Linear variables
in one variable: A linear equation in
one variable x is an equation of the form ax = b, where a and b are real
numbers, with a not equal to 0
Strategy for
Solving a Linear Equation
1. If fractions are present, multiply each side
by the LCD to eliminate them. If
decimals are present, multiply each side by a power of 10 to eliminate them.
2. Use the distribute property to remove
parentheses.
3. Combine any like terms.
4. Use the addition property of equality to get
all variables on one side and numbers in the other side.
5. Use the multiplicative property of equality
to get a single variable on one side.
6. Check by replacing the variable in the
original equations with your solution.
Formula/literal
equation: an equation involving two or
more variables, i.e., A = LW expresses the relationship among length L, width
W, and area A of a rectangle
Function: a function is a rule for determining uniquely
the value of one variable a from the value(s) of one or more other
variables. We say that a is the function
of the other variable(s)
Interest
rate: I = Prt
Trapezoid: A = 1/2h(b^1 + b^2)
Volume of
rectangular solid: V = LWH
Summary: Verbal Phrases and Algebraic Expressions
Addition: The sum of a number and 8 x + 8
Five is added to a number x + 5
Two more than a number x + 2
A number increased by 3 x + 3
Subtraction: 4 is subtracted from a number x - 4
3 less than a number x - 3
The difference between 7
and a number 7 - x
A number less 5 x - 5
Multiplication: The product of 5 and a number 5x
7 times a number 7x
Twice a number 2x
One half a number 1/2x (or x/2)
Division: The ratio of a number to 6 x/6
The quotient of 5 and a
number 5/x
3 divided by some number 3/x
Strategy for
solving word problems
1. Read the problem until you understand the
problem. Making a guess and checking it
will help you to understand the problem.
2. If possible, draw a diagram to illustrate the
problem.
3. Choose a variable and write down what it
represents.
4. The present any other unknowns in terms of
that variable.
5. Write an equation that models the situation.
6. Solve the equation.
7. Be sure that your solution answers the
question posed in the original problem.
8. Check your answer by using it to solve the
original problem (not the equation).
Geometic problems: Any problem that involves a geometric figure.
Uniform motion
problems: problems that involve motion
at a constant rate (D = RT: distance = rate x time).
Commission: when property is sold, the percentage of the
selling price that the selling agent receives
Inequality of
symbols
< is less than
< is less than or equal to
> is greater than
> is greater than or equal to
Solution
set: the set of all such numbers to an
inequality
Basic interval
notation (k any real number)
x > k (k, infinity symbol)
x >
k [k, infinity symbol)
x < k (- infinity symbol, k)
x <
k (- infinity symbol, k]
Linear
inequality
A linear
inequality in one variable x is any inequality of the form ax < b, where a
and b are real numbers, with a not equal to 0.
In place of < we may also use <, >, or >.
Properties of
Inequality
Addition
Property of Inequality: if the same
number is added to both sides of an inequality, then the solution set to the
inequality is unchanged
Multiplication
Property of Inequality: if both sides of
an inequality are multiplied by the same positive numbers, then the solution
set to the inequalilty is unchanged. If
both sides of an inequality are multiplied by the same negative number, and the
inequality symbol is reversed, then the solution set ito the inequality is
unchanged.
Equivalent
inequalities: inequalities with the same
solution set
Verbal sentence
Inequality
x is greate than
6; x is more than 6 x > 6
y is smaller
than 0; y is less than 0 y < 0
w is at least
9; w is not less than 9 w > 9
m is at most
7; m is not greater than 7 m < 7
Trichotomy
property: three possible ways to
position two real numbers on a number line
Trichotomy
Property: for any two real numbers a and
b, exactly one of these is true: a <
b, a = b, or a > b.
Simple
inequalities: refer to inequalities
Compound
inequalities: when joining two simple
inequalities with the connective "and" or the connective
"or." A compound inequality
using the connective "and" is ture if and only if both simple
inequalities are true. If at least one
of the simple inequalities is false, then the compound inequality is false.
Summary of Basic
Absolute Value Equations
Absolute Value
Equation Equivalent Equations Solution Set
|x| = k (k >
0) k = k or x =
-k {k, -k}
|x| = 0 x =
0 {0}
|x| = k (k <
0) x = 0 empty set
Summary of Basic
Absolute Value Inequalities (k > 0)
Absolute Equivalent Solution Graph of
Value Equality Set Solution Set
Inequality
|x| > k x > k or x < -k (-infinity, k)u(k, infinity)
|x| >
k x > k or x <
-k (-infinity k]u[k, infinity)
|x| < k x < k or x > k (-k, k)
|x| <
k x < k or x <
k [-k, k]
x-axis: the horizontal number line
y-axis: the vertical number line
Cartesian
coordinates system: every point in the
plane corresponds to a pair of numbers -- its location with respect to x-axis
and its location with respect to the y-axis
Rectangular
coordinate system: also refers to
Cartesian coordinate system
Origin: the intersection of the axis
The axes divide
the coordinate plane or xy plane into four regions called quadrants
Locating a point
in the the xy-plane that corresponds to a p air of real numbers is referred to
as plotting or graphing the point
A pair of
numbers, such as (2, 4) is called an ordered pair because the order of
the numbers is important
The first number
in an ordered piar is the x-coordinate and the second number is the y-coordinate
LINEAR EQUATION
IN TWO VARIABLES
A linear
equation in two variables is an equation of the form Ax + By = C, where A
and B are not both zero
Since the value
of y in y = 2x + 3 is determined by the value of x, y is a function of x. Because the graph of y = 2x + 3 is a line,
the equation is a linear equation and y is a linear function
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