A letter

that represents

a number.

Algebraic Expression

(Arithmetic expression)

1. Algebraic Expression is a "bundle" of:
   • variable(s) / number(s)
   • mathematical operations 
   • addition, 
   • subtraction,
   •  multiplication, 
   • division 
2. Arithmetic Expressions have no variables.

equation that

shows relationship

between variables

"divisibility test"

of

4

divisible by 4 if

the number formed by 

it's last two digits

is divisible by 4.

"divisibility test" of 5

"divisibility test"

of

8

if the number formed by 

it's last three digits

is divisible by 8

factor

 divisor

A natural number that

divides into itself

with no remainder.

• natural number = 12 
• factors of 12 = 1, 2, 3, 4, 6, 12

A whole number greater than 1 that has exactly two divisors; 1 and itself.

{ex; 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...}

4, 6, 8, 9, 10, 12,...

• whole number
• greater than 1
• NOT a prime number

 A 

 B

• A & B = whole numbers 
• B ≠ 0

A whole number and it's opposite. 

{..., -3, -2, -1, 0, 1, 2, 3, ...}

A statement that a quantity is

(< less than, ≤ less than or equal to, > greater than, ≥ greater than or equal to or ≠ )

not equal to another quantity.

Two numbers that are the same distance from zero (on a number line) but on opposite sides.

{0 is it's own opposite}

• Any number that can be written in the form p/q
• p, q = integers
• q ≠ 0

... √2 ... Π ... e ...

• nonterminating
• non-repeating
• decimal
• can't write in p/q form

Any number that is

anywhere on the number line.  

Rational and rational numbers combined.

A number which has the square root of a whole number.

{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}

A number or polynomial which is the exact cube of another number or polynomial.

{1, 8, 27, 64, 125}

Changing the order when adding does not affect the answer.  For any real numbers a and b;

a + b = b + a

{commutative, adj moving around, back and forth}

Changing the grouping when adding does not affect the answer.  For any real numbers a, b, and c;

(a + b) + c = a + (b + c)

{associate, to join a group}

When 0 is added to any real number, the result is the same real number.  For any number a;

a + 0 = a and 0 + a = a

addition property of opposites

AKA

inverse property of addition

 a + (-a) = 0

The sum of

a number and it's 

opposite (additive inverse) 

is 0.

• a = real number a 
• -a = opposite or additive inverse

 In subtraction

50 - 16 = 34

50 is the minuend

A quantity or number that is to be subtracted from another.

{ex; in the equation 50 - 16 = 34, the subtrahend is 16}

range

of a set

of numbers

R = H - L

• R=range
• H=high value 
• L=low value.

pos. x neg.

or

neg. x pos.

Changing the order when multiplying does not affect the answer.  For any real numbers a and b; 

ab = ba

Changing the grouping when multiplying does not affect the answer.  For any real numbers a, b, and c; 

(ab)c = a(bc)

The product of 0 and any real number is 0.  For any real number a;

0 · a = 0 and a · 0 = 0

multiplication property of 1

AKA

identity property of multiplication

1 * a = a               a * 1 = a

The product of 1

and any number

is that number

• a=real number

inverse property

of multiplication

a * (1/a) = 1

• a = number 
• (1/a) = multiplicative inverse, reciprocal 

division properties 

of 1 and 0

• a=real number≠0
• Any number divided by 1 is the number itself.  

a/1 = a 

• Any number divided by itself is 1.  

a/a = 1

3⁵ = 3·3·3·3·3

• exponent = repeated multiplication
• base = the factor being multiplied

(-4)²

vs.

-4²

• negative 4 squared

(-4)² = (-4) * (-4) = 16

• the opposite of 4 squared

-4² = - (4·4) = -16

Please - parentheses (all grouping symbols)

Excuse - exponents (and roots)

My Dear - mult./div. (from left to right)

Aunt Sally - add./sub. (from left to right)

To find the mean of a set of values:

1. divide the sum of the values 
2. by the number of values

pieces of equation

separated by addition.

 6x² + 3x - 8y 

The numerical factor of a term.

6x² + 3 -(1)y

Terms with exactly the same variables raised to exactly the same powers. Constant terms are always considered to be like terms.

{ex; 8x² and -1/2x² are like terms, 

8x² and -1/2x are NOT like terms}

8(4x) = 32x

Two expressions 

one variable

same value 

For any real numbers a, b and c;

a(b + c) = ab + ac

{read as; a times the quantity of b plus c}

sum of opposites

- ( a + b ) = -a + (-b)

Like terms can be combined by adding or subtracting the coefficients of the terms and keeping the same variable with the same exponent.

{ex; 2x + 9x + 3x² = 11x + 3x²}

A number that makes an equation true when substituted for the variable.  It is said to "satisfy" the equation.

The solution set of an equation is the set of all numbers that make the equation true.

addition

property of

equations and inequalities

 a = b               a + c = b + c

add the same number

to both sides

and get an equivalent equation.

Subtraction

property of

equations

a = b               a - c = b - c

subtract same number

from both sides

get an equivalent equation

• a, b, c = real numbers

multiplication property

of

equations and equalities

 a = b          ca = cb

multiply both sides of equation

by the same nonzero number

to get equivalent equation.

division property of

equations and equalities

(a = b) = (a/c = b/c)

Divide both sides

by same nonzero number

and get equivalent equation.

symmetric

property of

equations

interchange the sides of an equation,

to get an equivalent equation. 

fraction of  X  =  X/1

M Inverse of  X  =  1/X

An equation with no solution.

0*x = 11                      x + 1 = x + 9

:(                                     :(

ø

AKA

null set

AKA

empty set

The set of

an equation

with no solutions

x = x + 1

:(

An equation in which

every number is a solution.

x*0 = 0                      x + x = 2x

Where a, b and c are real numbers and a ≠ 0;

ax + b = c

LCD

least common denominator

least common multiple of the denominators

smallest whole number

of denominator(s)

divisible by all denominators.

clearing fractions  

in an equation

1/2 + 2/3x = -1/4

1) find LCD; 

1/2 + 2/3x = -1/4            LCD OF (2 , 3 & 4) =12

2) clear fractions;

12 (1/2 + 2/3x) = 12 (-1/4)

12 ₆ · 1/2 + 12 4 · 2/3x = 12 ₃ · -1/4

6 + 8x = -3

3) isolate variable;

6 (-6) + 8x = -3 (-6)

(8x) ÷ 8 = (-9) ÷ 8

x = -9/8

4) check;

1/2 + 2/3(-9/8) = -1/4 ?

1/2 + -3/4 = -1/4 ?

-1/4 = -1/4 :)

1/100 of an object.

• per = out of
• cent = 100.  

how to convert;

fraction -> decimal -> percent

1) fraction to decimal, 

numerator ÷ denominator = decimal

{ex; 11/20 = 11 ÷ 20 = 0.55}

2) decimal to percent,

move decimal point two places to the right.

{ex; 0.55 = 055. = 55%}

how to convert;

percent -> decimal -> fraction

1) percent to decimal, 

move the decimal point two places to the left.

{ex; 15% = 15. = 0.15}

2) decimal to fraction,

write the decimal number in fraction form, then reduce if possible.

{ex; 0.15 = 15/100 = 3/20}

X  is  "what" %  of  Y

The reduction in price

of an item

expressed as a percentage

percent change

...

percent increase

...

percent decrease

% change  =  new value - original value   × 100
       original value

% efficiency = output · 100

               input

% concentration =     amount of A       ·  100%

                                  amount of mixture

An equation that

states a relationship

between two or more variables.

r = c + m

• r  =retail price
• c  =cost to merchant
• m =markup

p = r - c

• P = profit
• R = revenue 
• C = cost

interest = principal · rate · time

i = p · r · t

formula for

distance traveled

d = r · t

• d=distance
• r=rate
• t=time

Formula for converting

Celsius to Fahrenheit 

F = (9/5)C + 32

area equals side squared

A = s²

A = l · w

• A = area 
• L = length 
• W = width

area

of a

parallelogram

A = b · h

• A=area
• B=base
• H=height

Area equals one half (times) base times height

A = 1/2 · b · h

A = 1/2 · (b₁ + b₂) · h

• A=area
• b₁=base 1 
• b₂=base 2 
• H=height.

area equals pi times radius squared.

A = π · r²

perimeter equals two times the length plus two times the width.

P = 2l + 2w

perimeter equals four times one side.

P = 4s

Perimeter equals side A plus side B plus side C.

P = a + b + c

diameter is twice the radius.  

D = 2r

r = 1/2D

• R=radius
• D=diameter

C = 2πr

• C=circumference
• R=radius
• π=pi 

C = πD

• D=diameter

V = l · w · h

• V=volume
• l=length
• w=width
• h=height.

V = s³

•  V=volume
•  S=side

V = 4/3 · π · r³

• V=volume
• Π=pi
• r³=radius cubed.

V = π · r² · h

• V =volume
• π =pi
• r² =radius squared
• h =height.

V = 1/3 · π · r² ·h

• V=volume
• π=pi
• r²⁼radius squared
• H=height.

V = 1/3 · B · h

• V=volume
• B=base
• H=height.

1. Write down EXACT question
2. a for base unknownPick variable
3. form equation
4. Solve equation
5. State solution in a complete sentence
6. check

measure of a

right angle

sum of

the measure of

2 complimentary angles

Two equal sides and two equal angles.

[image]

BASE ANGLES

of an isosceles triangle

two equal/congruent angles of the triangle.

[image]

vertex angle

(of an isosceles triangle)

angle that is not equal to the other two angles.

[image]

linear inequality

in one variable

ax + b > cax + b ≥ c

ax + b < cax + b ≤ c

• a, b, c = real numbers 
• a≠0
• x = variable

set-builder

notation

{xlx > -3}

"the set of all x such that x is greater than -3"

describes solution set of inequality

when there are too many solutions to list.

HOW TO GRAPH

an inequality

• Draw a picture on a number line 
• Line = all numbers that make inequality true. 
• (-∞,21/2] : 

[image]

parenthesis "()" or open circle "ο"

Inequality graphing symbols that indicate the number is not part of the graph.

Inequality graphing symbols

...

that indicate

...

number is part of the graph.

Intervals written in compact form.

{ex; [ -3, ∞ ) is the interval notation that represents all solutions of the variable of an inequality that includes -3 and continues on (to the right/pos. direction) forever, but never reaches infinity.}

addition and subtraction

properties

of inequality

Adding the same number to,

or subtracting from,

both sides of an inequality

does not change its solutions.  

(a < b) = (a + c < b + c)

(a < b) = (a - c < b - c)

multiplication and division

properties of inequalities

• Multiply or divide both sides by a POSITIVE#

NO CHANGE

• Multiply or divide both sides by a NEGATIVE#

inequality symbol must reverse direction

• -2 < x < 6
• -2 < x
• x < 6

• (-2,6) 
• solutions: -1, 0, 1, 2, 3, 4, 5

The point where

x-axis and y-axis

intersect.  

(0,0)

Top right

[image]

Top left

[image]

Lower left

[image]

Lower right

[image]

Each point in a coordinate plane can be identified by an ordered pair of real numbers x and y written in the form (x,y).

NOTE: An ordered pair must be written with parentheses;

x,y :(

(x,y) :)

The first number, x, in an ordered pair.

(x,y)

The second number, y, in an ordered pair. 

(x,y)

The process of locating a point in the coordinate plane.

point "P" with the ordered pair (4,2);

1) Start at origin.

2) plot X-coordinate first, move 4 units right.

3) plot y-coordinate next, move 2 units up.

4) make a dot and mark it with letter and ordered pair.

[image]

A process that

uses known information

to predict values

that are not known

but are within the range of data.

Using known information

...

to predict values

...

outside range of data

solution

of an equation

in two variables

an ordered pair (x,y)

LINEAR EQUATION

in two variables

Ax + By = C

• An equation 
• Variables = X , Y
• Raised to FIRST power   
• A, B, C = real numbers 
• A, B ≠ 0

To draw a picture of

---
ALL the solutions

...
of the equation.

Any point where graph touches  x-axis.

[image]

Any point of a graph where the graph touches/crosses the y-axis.

[image]

1. substitute 0 for y in equation 
2. solve for x.

A horizontal line always has the equation of y=a where a is a fixed number.

ex; y=7

[image]

x=b

[image]

• variable = m 
• measures a lines: 
• steepness
• orientation
• left to right:
• uphill = positive slope 
• downhill = negative slope

• bigger slope = steeper line.

slope = rise =   vertical change  = change in y's = Δy

             run     horizontal change    change in x's    Δx

[image]

slope of

a horizontal line

• M = 0 over Y 
• 0 over Y = 0 
• 0 = "zero slope"

[image]

SLOPE FORMULA

with 2 points

m = Y₂ - Y₁

       X₂ - X₁

Line goes through 2 points

(X₁,Y₁) and (X₂,Y₂)

• Two lines 
• cross/intersect at a 90° angle
• slopes are opposite reciprocals.
• m₁= 2 / 3 
• m₂= - 3 / 2

subscript notation is to demonstrate that there is more than one number for the variables a and b, therefore distinguishing between them.

a₁, a_2, b₁, b₂

read; "a sub 1, a sub 2, b sub 1, b sub 2"

slope-intercept equation

of a line

y = Mx + B

• M = slope
• B= (0,B), y-intercept

point-slope equation

of a line

y - y₁ = m(x-x₁)

• point = (x₁,y₁)
• slope = M

 Regions of

a coordinate plane

bound by a line

A line that divides the coordinate plane

into two half-planes

after solving for the equation given.  

• solid line with  ≤  ≥
• dotted line with  <  > 

Graphing linear inequalities

in two variables.

1. Replace < > ≤ ≥ with = 
2. Graph boundary line of the region.  
1. Dotted line for < > 
2. solid line for ≤ ≥
3. Pick a test point not on the boundary line.  
4. Use coordinates of test point to solve.
1. true - shade that side.
2. false - shade other side.

( FIRST, SECOND )

Components

of ordered pairs.

• Ordered pair = ( X , Y )
• X = FIRST component
• Y = SECOND component

A set of ordered pairs within {}

{(84,8), (88,6), (92,11), (94, 13)}

The set of first components of a relation.

{(84,8), (88,6), (92,11), (94, 13)}

The set of

all second components

of a relation.

{(84,8), (88,6), (92,11), (94,13)}

ƒ(X)=Y

• X=domain
• Y=range

• X = INPUT : The starting variable

• Y = OUTPUT : The resulting variable

The notation y = ƒ(x)

denotes that the variable y is a function of x.

A function in which the graph is a non-vertical line.

ex; ƒ(x) = 4x + 1

Absolute value function.

independent

and

dependent

variables

If y depends on x,

y = DEPENDENT variable

x = INDEPENDENT variable

find y-intercept

1. substitute 0 for x in equation 
2. solve for y.

How to solve

a linear equation

in two variables

1. pick number for x 
2. solve for y
3. solution is ordered pair
4. check solution

Formula for converting

Fahrenheit to Celsius

C = 5/9 (F - 32)

The formation of two equations with the same variables considered simultaneously.

{imagine bracket encompassing both equations here;

x+y=3

x-y=1 

read as "the system of equations x + y = 3 and x - y = 1"

A system of equations that has at least one solution.

ex;

[image]

A system of equations with NO solution.

ex;

[image]

Equations with different graphs.

ex;

[image]

and

[image]

are both independent equations

Essentially the same equation graphed twice.  In this case, there are infinitely many solutions to dependent equations.

ex;

[image]

1) Carefully graph each equation on the same rectangular coordinate system.

2) If the lines intersect, determine the coordinates of the point of the intersection of the graphs.  That ordered pair is the solution of the system.

3) Check the proposed solution in each equation of the original system.

[image]

An algebraic method for solving a system of equations.

1) Solve on of the equations for either x or y.  If this is already done, go to step 2. (<=substitution equation)

2) Substitute the expression for x or for y obtained in step 1 into the other equation and solve that equation.

3) Substitute the value of the variable found in step 2 into the substitution equation to find the value of the remaining variable.

4) Check the proposed solution in each equation of the original system.  Write the solution as an ordered pair.

ex;

[image]

A method for solving a system based on the addition property of equality.

1) Write both equations of the system in standard

Ax + By = C form.

2) If necessary, multiply one or both of the equations by a nonzero number chosen to make the coefficients of x (or the coefficients of y) opposites.

3) Add the equations to eliminate the terms involving x (or y).

4) Solve the equation resulting from step 3.

5) Find the value of the remaining variable by substituting the solution found in step 4 into any equation containing both variables.  Or repeat steps 2-4 to eliminate the other variable.

6) Check the proposed solution in each equation of the original system.  Write the solution as an ordered pair.

ex;

http://www.youtube.com/watch?v=awIMaSkY_g4

(can't find a clear image of this as an example, method can be reviewed with this video if necessary.)

A) You can write each equation in standard (Ax + By = C) form and use elimination.

B) You can solve for the variable with the coefficient of 1 or -1 and use substitution.

1) You see the solutions.

2) The graphs allow you to observe trends.

1) Always gives the exact solutions.

2)Works well if one of the equations is solved for one of the variables, or if it is easy to solve for one of the variables.

1) Always gives the exact solutions.

2) Works well if no variable has a coefficient of 1 or -1.

1) Inaccurate when the solutions are not integers or are large numbers off the graph.

2) This method can be lengthy.

1) You do not see the solution.

2) If no variable has a coefficient of 1 or -1, solving for one of the variables often involves fractions.

1) You do not see the solution.

2) The equations must be written in the form

Ax + Bx = C.

1) Analyze the problem by reading it carefully to understand the given facts.  Often a diagram or table will help you visualize the facts of the problem.

2) Pick different variables that represent two unknown quantities.  Translate the words of the problem to form two equations involving each of the two variables.

3) Solve the system of equations using graphing, substitution or elimination.

4) State the conclusion.

5) Check the results in the words of the problem.

Amount of liquid*Strength of solution=amount of alcohol

Ex;

8 (oz) * 0.55 (55%) = 4.4 (440%)

Amount * Price = Total value

Ex;

10 items * $5.00 = $50.00 value

Rate * Time = Distance

Ex;

25 mph * 2 hours = 50 miles.

To find an ordered pair that satisfies each inequality.

1) Graph each inequality in the same rectangular coordinate system. (Using; "the intercept method", the "slope and y-intercept method" or "A table of solutions".)

2) Use shading to highlight the intersection of the graphs ( the region where the graphs overlap). The points in this region are the solutions of the system.

3) As an informal check, pick a pint from the region where the graphs intersect and verify that its coordinates satisfy each inequality of the original system.

Ex; Graph the system

[image]

[image]

"must be at least"

"cannot go below"

"is not more than"

"should not surpass"

A Linear Equation in Three Variables

(standard/general form)

An equation that can be written in the form;

Ax + By + Cz = D

Where A, B, C, and D are real numbers and A, B, and C are not all 0.

An ordered triple whose coordinates satisfy the equation.  Ex;

(2,0,1) is a solution of x + y + z = 3

An ordered triple that satisfies each equation of the system.

Ex;

(-4,2,5) is a solution of

   {2x + 3y + 4z = 18

{3x + 4y + z = 1

{x + y + 3z = 13

1) Write each equation in standard form Ax + By + Cz = D and clear any decimals or fractions. 

2)  Pick any two equations and eliminate a variable.

3)  Pick a different pair of equations and iliminate the same variable as in step 1.

4) Solve the resulting pair of two equations in two variables.

5)  To find the value of the third variable, substitute the values of the two variables found in step 4 into any equation containing all three variables and solve the equation.

6)  Check the proposed solution in all three of the original equations.  Write the solution as an ordered triple. 

The power of an expression.

In the expression x^2; 

2 is the exponent.

A number or expression to be raised to a power.

In the expression x^2;

x is the base.

Tell how many times its base is to be used as a factor.

For any number x and any natural number n;

xⁿ = x * x * x * x....

       ("n" factors of x)

• Expressions of the form Xⁿ. 

• The base of an exponential expression can be a number, a variable, or a combination of numbers and variables.  

• EX; 10⁵ = 10 * 10 * 10 * 10 * 10

The base is 10.  The exponent is 5.  

Read as "10 to the fifth power"

(NOTATION) 

writing bases that contain " - "

• These must be written within parentheses.

• ex;  "negative 2X raised to the third power" or "negative 2X cubed."

(-2x)³ = (-2x) * (-2x) * (-2x)

or -2*-2*-2*x*x*x

or (-2)³ * X³

-8X³

• However, without parentheses, "The opposite of 8 to the fourth power is";

-8⁴ = - (8*8*8*8)

-4096

To divide exponential expressions that have the same base, keep the common base and subtract the exponents.

For any nonzero number X and any natural numbers M and N, where M>N;

X^M

---- = X^M-N

X^N

"X to the Mth power divided by X to the Nth power equals X to the M minus Nth power."

DO NOT DIVIDE THE BASES, USE THE SAME BASE.

• To raise an exponential expression to a power, keep the base and multiply the exponents.
• For any number x and any natural numbers m & n;

(x^m)ⁿ = x^m*n = x^mn

• "The quantity of x to the m^th power raised to the n^th power equals x to the mn^th power."

• To raise a product to a power, raise each factor of the product to that power.  
• For any numbers x and y, and any natural number n;

(xy)ⁿ = xⁿyⁿ , where y ≠ 0

If m and n represent natural numbers and there are no divisions by zero, then;

x¹ = x

• Any nonzero base raised to the 0 power is 1.
• For any nonzero real number x;

x⁰ = 1

• (0⁰ = indeterminate form)

• For any nonzero number x and any integer n;

x^-n = 1/xⁿ

• In words, "x^-n is the reciprocal of x"

• A factor can be moved from the denominator to the numerator (or vice versa) of a fraction if the sign of its exponent is changed.  
• For any nonzero real numbers x and y, and any integers m and n;
• 1/x^-n = xⁿ
• x^-m/y^-n = yⁿ/x^m
• -5s^-2/t^-9 =-5t⁹/s² ≠ t⁹/5s²

• A fraction raised to a power is equal to the reciprocal of the fraction raised to the opposite power.
• For any nonzero real numbers x and y, and any integer n;

(x/y)^-n = (y/x)ⁿ

1. If the exponent is positive, move the decimal point the same number of places to the right as the exponent.
2. If the exponent is negative, move the decimal point the same number of places to the left as the (absolute) value of the exponent.

Polynomial Etymology

(not in book)

• Etymology;  poly- + -nomial.   polynomial (plural polynomials)

1. (algebra) An expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negativeinteger power, such as a_nxⁿ + a_{n − 1}x^{n − 1} + ... + a₀x⁰.

• POLY - 

From Ancient Greek (πολύς, polus)

 “many, much”

• NOMINAL 

From the (Middle English, nominalle)

 “of nouns” 
 

Latin (nōminālis, nōmen) 

 “name”.

• Incorrect etymological progression.  Should have been "polynominal"

• A single term or a sum of terms in which all variables have whole-number exponents and no variable appears in a denominator.  Ex; 

• 4y² - 2y - 3  
• a³ + 3a²b + 3ab² + b³

• The expression 6x³ + 4x^-2 is not a polynomial because of the negative exponent on the variable in the term "4x^-2"
• The expression y² + 5/y + 1 is not a polynomial because of the variable "y" appearing in the denominator in the term "5/y"

Polynomial in one variable, x;

3x +2  
"the sum of two terms 3x and 2."

• A single number.
• In 3x +2; "2" is the constant term
  

• To descend; to go or move downward, to lower.

• The exponents of the variable (x) are written from left to right.

• 4x² + (-2X) + (-3)

(x², x¹, x⁰)

• The first term in a polynomial.
• 4x² +(-2X) + (-3)

• The coefficient of the leading term.
• 4x² +(-2X) + (-3)

• To ascend; To go or move upward, rise.

• The exponents of the variable "x" in a polynomial increase from left to right.

• 4 + (-2X) + (-3x²)

(x⁰, x¹, x²)

• Three variables; x, y and z.
• One term.
• -8xy²z 

• Mono - From Ancient Greek "μόνος - monos" - alone, only, sole, single.
• A polynomial with exactly ONE term.