Ch. 1, Sec. 1
Notes � Variables are letters that stand in place of an unknown number. � Equations always have an equal sign. � More means to add. Less means to subtract. Product means to multiply. Quotient means to divide.
Ch.1, Sec.2
Notes � An exponent is a number that tells how many times to multiply the base by itself. � Simplify means to solve, reduce, or put in simplest form. � X to the second power is the same as X times X, or X^2 or X squared. � P: Parenthesis E: Exponent M: Multiply D: Divide A: Add S: Subtract
Ch.1, Sec.3
Notes � Natural Numbers: 1,2,3, N � Whole Numbers: 0,1,2,3, N � Integers: N, 2, 1, 0, 1, 2 N � Rational Numbers: A/B where B is not equal to Zero. � Terminating Decimals are decimals that end. � Repeating Decimals are decimals that repeat and go on forever. � Rational Numbers are numbers that either terminate or repeat. � Irrational numbers are numbers that go on forever, but don�t repeat. � A counterexample is an example that proves a statement false. � Equalities have an equal sign. � Inequalities have either a symbol or a symbol. � When ordering fractions, the larger the denominator, the smaller the value of the fraction. � When dividing fractions, you multiply by the reciprocal. � The absolute value of a number is its distance from zero on a number line.
Ch.1, Sec. 4
Notes When multiplying/dividing positive and negative numbers: � If both numbers are positive, then the answer is positive � If both numbers are negative, then the answer is negative. � If the signs are different, then the answer is negative. � Evaluate means to substitute a variable. +1 + 1 2 -1 + 1 0 -1 � 1 -2 +1 � 1 0
Ch.1, Sec.6
Notes � Identity property: Anything multiplied or divided by 1 is equal to itself. � Zero Property: Anything multiplied or divided by 0 is equal to zero. � Anything multiplied by �1 is equal to that numbers opposite. � When multiplying numbers with the same sign, the answer will always be positive. � When multiplying numbers with different signs, the answer will always be negative. � Inverse property: Anything multiplied by its multiplicative inverse is equal to one.
Ch.1, Sec.7
Notes The Distributive Property 4(a+b) 4a+4b 3(2x+5) = 6x +15 x(2x+3) = 2x+3x = 5x
Ch.1, Sec. 8
Notes � Commutative Property of Addition/Multiplication: You can switch the order in which you add/multiply the integers, without changing the sum/product. 3+7 is the same as 7+3 � Associative Property of Addition/Multiplication: You can change the grouping of the integers without changing the sum/product. (2x3) X 5 is the same as 2(3x5) � Identity Property of: Addition-anything added to zero is equal to itself. Multiplication-anything multiplied by 1 is equal to itself. � Inverse Property of: Addition- anything added to its opposite is equal to zero. Multiplication-anything multiplied by its inverse is equal to 1. � Multiplication Property of �1 N x �1 = -N
Ch.1, Sec.9
Notes (2, 3) II I III IV An ordered pair shows something�s location on a coordinate plane. Each of the 4 sections is called a quadrant.
Ch.2, Sec. 1
Notes 3 + X =10 -3 -3 X =7 -1 + 3 + X �2X = 10 -1 + 3 � X = 10 2 - X = 10 -2 -2 -X = 8 x �1 x -1 X= -8 Always write work out to the side.
Ch.2, Sec.2
2x - 12= 4 given +12 +12 + 12 to both sides 2x = 16 simplify 2 2 divide both sides by 2 X = 8
Ch. 2, Sec.3
2c+c+12=78 3c+12=78 -12 -12 3c=78 3 3 C=26
Ch.2, Sec.4
2x+3=2+x To solve this, you must get X�s on one side and the integers on the other. -x -x x+3=2 -3 -3 x = -1
Ch.2, Sec.5
Jacksonville to Daytona = X Jacksonville to Gainesville=D X + D = the distance from Gainesville to Daytona Consecutive Integers are integers that are in numerical order. 1,2,3 To find consecutive integers: X + X+1 + X+2 3X + 3 Distance equals rate times time D = R x T To find same direction travel: 1. Set up the equations and set them as equal. 2. Solve, subtract beginning time, etc. To find round trip distances: 1. Set up the equations, set them as equal 2. Solve
Chapter 3-1
Inequalities and their Graphs x<5 2<5 so x can = 2 0 = just lower or greater than . = equal to and lower or greater than Solution of an inequality - any number that makes the inequality true.
Chapter 3-2
Solving Inequalities Using Additional and Subtraction x-3 >5 +3 +3 x>8 x+3 >5 -3 -3 x>2 Equivalent Inequalities- are inequalities with the same solutions.
Chapter 3-3
Solving Inequalities using multiplication and Division x 2 < -1 .2 .2 x<-2 -2x + 3+x > 6 -x + 3 > 6 -3 -3 -x > 3 .-1 .-1 x <-3
Chapter 3-4
Solving Multi- Step Inequalities 7 + 6a =9 -7 -7 6a = 2 6 6 a = .3333 7 + 6a > 9 -7 -7 6a > 2 6 6 a > .3333 -3(4-m) > 4(2m + 1) -12 + 3m > 8m + 4 +12 +12 3m > 8m + 16 -8m -8m -5m > 16 -5 -5 m > -3 1/5
Chapter 3-5
Compound Inequalities
Less than = and
greater than or
for and
-3 < j +2 < 7
-5
Chapter
3-6
Absolute Value Equations and Inequalities
[x] = 2
[x] + 5 = 11
[x] = 6
x = 6 or -6
and = greater than >
or = less than <
Chapter
4-1
Ratio and Proportion
20 Girls
6 Boys
girls boys
20:6
Unit Rate:
10 pcs. $75
10 = $7.50
a/b = c/d
means - extreme
ad = bc
cross multiplication
t/9 = 5/6
6t = 45
t = 7.5
Proportion - an equation that states that two ratios are equal.
Extremes of the proportion - for the equation a/b = c/d and d are the extremes.
Means of the proportion - for the same equation b and c are the means.
Cross products - are the ones you cross in and multiply. for a/b == c/d ad and bc are cross products.
Chapter
4-2
Proportions and Similar Figures
Corresponding sides: are having the ratio or proportion and there are 2 equal angles.
(~)Means they are similar
2500sq ft
With two sides 50ft
The scale is 10' to 1"
Similar figures-the same shape but not necessarily the same size.
Scale drawing-an enlarged or reduced drawing that is similar to an actual object or place.
Scale-ratio of a distance in the drawing to the corresponding actual distance.
Chapter
4-3
Proportions and Percent Equations
n or x
x or n x or n %
What percent of 80 is 18?�
N = 18
100 80
80n=1800 n=22.5
CENTER> ADDITIONAL:
Chapter 3 Notes
3-1
Inequalities and their Graphs
x<5
2<5 so x can = 2
<-------------0
<--------------------------------5--->
0 = just lower or greater than
. = equal to and lower or greater than
Solution of an inequality - any number that makes the inequality true.
3-2
Solving Inequalities Using Addition and Subtraction
x-3 >5
+3 +3
x>8
x+3 >5
-3 -3
x>2
Equivalent Inequalities- are inequalities with the same solutions.
3-3
Solving Inequalities using multiplication and Division
x
2 < -1
.2 .2
x<-2
-2x + 3+x > 6
-x + 3 > 6
-3 -3
-x > 3
.-1 .-1
x <-3
3-4
Solving Multi- Step Inequalities
7 + 6a =9
-7 -7
6a = 2
6 6
a = .3333
7 + 6a > 9
-7 -7
6a > 2
6 6
a > .3333
-3(4-m) > 4(2m + 1)
-12 + 3m > 8m + 4
+12 +12
3m > 8m + 16
-8m -8m
-5m > 16
-5 -5
m > -3 1/5
3-5
Compound Inequalities
Less than = and
Greater than or
For and
-3 < j +2 < 7
-5
Chapter 4 Notes
4-1
Ratio and Proportion
girls boys
20 6
20:6 girls
ratio is 26
girls boys
20 6
26 26
Unit rate
10 pcs. $75
75
10 = $7.50
a/b = c/d
Means - extreme
ad = bc
Cross multiplication
t/9 = 5/6
6t = 45
t = 7.5
Proportion - an equation that states that two ratios are equal.
Extremes of the proportion - for the equation a/b = c/d and d are the extremes.
Means of the proportion - for the same equation b and c are the means.
Cross products - are the ones you cross in and multiply. For a/b == c/d ad and bc are cross products.
4-2
Proportions and Similar Figures
Corresponding sides: are having the ration or proportion and there are 2 equal angles
.
. . Means they are similar
2500sq ft
with two sides 50ft
The scale is 10' to 1"
Similar figures - the same shape but not necessarily the same size.
Scale drawing - an enlarged or reduced drawing that is similar to an actual object or place.
Scale - ration of a distance into he drawing to the corresponding actual distance.
4-3
Proportions and Percent Equations
n or x
x or n x or n %
100
25 75
100 = 25% 320 = 23%
80 is 25% of what number
25 80
100 x
x = 320
4-4
Percent of Change
% triangle = change
New - old
old
If the New is bigger than old = Increase
If the Old is bigger than New = Decrease
16 16.70
1 x 1/2 = .5 = percent error
.10 x 1/2 = .05 = percent error
16.75 x 1/2 = .005 = percent error
Percent of Change - the ration amount of change over original amount experessed as a percent.
Percent of Increase - when a value increase from its original amount.
Percent of Decrease - when a value decreases from its original amount.
Greatest Possible Error - in a measurement is one half of that measuring unit.
Percent Error - greatest possible error over measurement.
4-5
Applying Ratios to Probability
Probability - possibility
Outcome - end result
Event - activity
Theoretical probability - # of favorable outcome over # of possible outcome
6 1
24 = 2
Experimental probability?
1000 skateboards
992 are good
992
1000 are good
99.2% are good
.008% is bad
Probability - tells you how likely it is that something will occur.
Outcome - the result of a single trial.
Event - any outcome of group of outcomes.
Sample Space - is all of the possible outcomes.
Theoretical probability - # of favorable outcome over # of possible outcome.
Complement of an event - consists of all the outcomes not in the event.
Experimental probability - number of times an event occurs over number of times the experiment is done.
4-6
Probability of Compound Events
Compounded: more than one time
Independent:
1B 2x
20 20
100 100
1 1 1
5 x 5 = 25
Dependent:
1 1 1
5 99 = 495
Chapter 5
5-1
Relating Graphs
5-2
Relations and Functions
The first numbers in an ordered pair are the domains the second numbers are the ranges once put in ruder.
Domain - Independent/ x axis
Range - dependent/ y axis
function: for every value of "x" you have a unique and
only one value of "y".
Vertical Line test:
It's not a function if theres more than one point on one vertical line.
y = mx +b
m= slope
b = "y" intercept
f(x) = mx+b means it's already a function.
Relation - a set of ordered pairs.
Domain - a relation is the set of first coordinates in an ordered pair.
Range - the set of second coordinate in an ordered pair.
Function - a relation that assigns exactly one value in the range to each value in domains.
Vertical line test - a way to analyze the graph of the relation.
Function Rule - an equation that describes a function.
Function Notation - when you use f(x) to indicate the outputs.
5-3
Function Rules, Tables, and Graphs
Xs = Domain, Independent variables
Ys = Range, dependent variables
For every x there is only on unique value of Y.
y=mx+b
Slope = rise over run = y2 - y1 over x2 -x1
b = y intercept (where the line intersects the y axis)
y = .5x +3
To find the "y" intercept set x = 0
To find the 'x' intercept set y =0
y = .5x + 3
Set y = 0
0 = .5x + 3
x = -6
Absolute Value: Do the same as any other just follow absolute value rules.
Independent variable - inputs are values.
Dependent variable - outputs are the corresponding values.
5-4
Writing a Function Rule
y = mx +b
f(x) means it's a proven function
1) Graph the data
2) Find out where does it the "y" intercepts
3) Find the slope rise over run
4) Must take 2 consecutive points
5-5
Direct Variation
y = kx
y = mx
It is not a direct variation
y = k x
k = coefficient of x
Constant of variation
(x,y)
(2,3)
write an equation with the constant of variation
y = kx
3 = k2
Divide both by 2
k = 3/2
y = 3/2x
Direct variation - function in the form y = kx, where k doesn't equal 0.
Constant of variation - k is the coefficient of x.
5-6
Describing Number Patterns
A(n) = a + (n - 1)(d)
a = First term
n = term number
d = difference in the terms
Example
A(n) = -9 + (n - 1)(6)
find for fifth term
-9 + (5 - 1)(6)
-9 + (4)(6)
-9 + 24 = 15
Inductive Reasoning - making conclusions based on patterns you observe.
Conjecture - conclusion you reach by inductive reasoning.
Sequence - number pattern.
Term - each number in a sequence.
Arithmetic sequence - adding a fixed number to each previous term.
Common difference - fixed number.
