Tuesday, July 17, 2018

Algebra Notes II

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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