Chapter 1 VOCAB & NOTES

Solving Equations & Inequalities

variables

variables

symbols (usually letters) used to represent unknown quantities

expressions

a mathematical representation using numbers, variables, exponents, and operations

rational numbers

3 subsets: natural (N) {1, 2, 3, …}, whole (W) {0, 1, 2, 3, …}, integer (Z or J) {…-3, -2, -1, 0, 1, 2, 3, …}

Terminating and repeating decimals can be written as a quotient of integers

irrational numbers

the set of non-repeating and non-termination decimals

commutative prop. (+/x)

switching order 5+4=4+5

associative prop. (+/x)

grouping/regrouping 4+7+x=(4+7)+x

distributive prop. (+/x)

3(x)+3(7) 3x+21 3(x=7)

inverse prop. (+)

3+(-3)=0 (opposite)

inverse prop. (x)

4x1/4=1 (reciprocal)

identity prop. (+)

7+0=7 add identity 0

identity prop. (x)

10x1=10 mult. identity 1

property of zero (x)

10x0=0

definition of =

1-7 1+-7

definition of div.

5/2 5x1/2

reflexive prop. (=) mirror

A=A x+7=x+7

symmetric prop. (=) switching

if A=B, then B=A 10=x+2 x+2=10

transitive prop. (=)

if A=B, and B=C, then A=C 3+4=7, 2+5=7 then 3+4=2+5

subtraction prop. (=)

x+4=10

-4 -4

addition prop. (=)

A=B, then A+C=B+C

multiplication prop. (=)

A=B, then AC=BC, C≠0

division prop. (=)

3x=12

/3 /3

set

a group of objects e.g. set of integers

element

each object within a set e.g. 2 E Z

empty set of null set

the set that contains no elements symbols: { }

The empty set is a subset of every set

subset ⊆

A is a subset of B if every element in A is also an element of B

Symbol: ⊆

Every set is a subset of itself

The empty set is a subset of every set

2^n = S

formula that represents relationship when n=# of elements and s=# of subsets

roster rule

lists members in set e.g. {1, 3, 5, 7, …}

literal rule

describes set e.g. {odd whole numbers}

algebraic rule

expression e.g. {2x-1|x⊆N} x=1, 2, 3, 4

union ∪

the union of A and B is a set C that contains all elements that are in either A or B

Put the two sets (A&B) together

intersection ∩

the intersection of A and B is a set C that contains all elements that in both A and B

Overlap or what two sets (A&B) only have in common

disjoint set

if A ∩ B = empty set

solution

a value that makes a sentence(s) true

conditional

one solution

contradiction

no solutions

identity

many solutions

transitive prop.

is a < b and b < c, then a < c

trichotomy prop.

if a and b are any real numbers then one of the following must be true a < b a > b a=b

set notation & interval notation

{x|x > 3} → (3, ∞) {x|x ≤ -4} → (-∞, -4\] {x|-8 < x < 1} → (-8, 1)

≤, ≥ → opens points on graph, \[ \]

<, > → closed points on graph, ( )

applications

coin problems, rate x time = distance

absolute value |n|

the distance from 0

conjunction

AND ∩ intersection means to share, in common

disjunction

OR ∪ union means to join

compound inequalities

Multiple inequalities

Scratchwork above number line

No intersection = no solution

Write answer in interval notation

CHAPTER 1 REVIEW

Evaluate expressions with variables

Use interval notation to solve inequalities

Name by letter all of the subsets or R which each is a member