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Inequalities
Chapter 1 VOCAB & NOTES
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Solving Equations & Inequalities
Algebra
Inequalities
10th
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45 Terms
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variables
symbols (usually letters) used to represent unknown quantities
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expressions
a mathematical representation using numbers, variables, exponents, and operations
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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
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irrational numbers
the set of non-repeating and non-termination decimals
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commutative prop. (+/x)
switching order 5+4=4+5
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associative prop. (+/x)
grouping/regrouping 4+7+x=(4+7)+x
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distributive prop. (+/x)
3(x)+3(7) 3x+21 3(x=7)
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inverse prop. (+)
3+(-3)=0 (opposite)
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inverse prop. (x)
4x1/4=1 (reciprocal)
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identity prop. (+)
7+0=7 add identity 0
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identity prop. (x)
10x1=10 mult. identity 1
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property of zero (x)
10x0=0
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definition of =
1-7 1+-7
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definition of div.
5/2 5x1/2
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reflexive prop. (=) mirror
A=A x+7=x+7
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symmetric prop. (=) switching
if A=B, then B=A 10=x+2 x+2=10
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transitive prop. (=)
if A=B, and B=C, then A=C 3+4=7, 2+5=7 then 3+4=2+5
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subtraction prop. (=)
x+4=10
-4 -4
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addition prop. (=)
A=B, then A+C=B+C
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multiplication prop. (=)
A=B, then AC=BC, C≠0
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division prop. (=)
3x=12
/3 /3
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set
a group of objects e.g. set of integers
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element
each object within a set e.g. 2 E Z
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empty set of null set
the set that contains no elements symbols: { }
The empty set is a subset of every set
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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
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2^n = S
formula that represents relationship when n=# of elements and s=# of subsets
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roster rule
lists members in set e.g. {1, 3, 5, 7, …}
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literal rule
describes set e.g. {odd whole numbers}
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algebraic rule
expression e.g. {2x-1|x⊆N} x=1, 2, 3, 4
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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
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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
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disjoint set
if A ∩ B = empty set
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solution
a value that makes a sentence(s) true
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conditional
one solution
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contradiction
no solutions
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identity
many solutions
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transitive prop.
is a < b and b < c, then a < c
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trichotomy prop.
if a and b are any real numbers then one of the following must be true a < b a > b a=b
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set notation & interval notation
{x|x > 3} → (3, ∞) {x|x ≤ -4} → (-∞, -4\] {x|-8 < x < 1} → (-8, 1)
≤, ≥ → opens points on graph, \[ \]
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applications
coin problems, rate x time = distance
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absolute value |n|
the distance from 0
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conjunction
AND ∩ intersection means to share, in common
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disjunction
OR ∪ union means to join
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compound inequalities
Multiple inequalities
Scratchwork above number line
No intersection = no solution
Write answer in interval notation
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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