Algebra 2 Chpt 2 Review
proof
A logical argument that shows a statement is true.
theorem
A mathematical statement which we can prove to be true
hypothesis
A testable prediction, often implied by a theory
Transitive Property of Inequality:
If a > b and b > c, then a > c.
dots (=)
Ix-3I=2/ x=x, a=3, d=2/ solution set: {1,5}
2 rays (>)
Iy+1I>4/ x=y, a=-2, d=4/ solution set: {y: y<-5 or y>3}/ interval notation: (-(infinity sign), -5) U (3, (+infinity sign))
Comparison Property:
One of the following must be true either a > b, a = b, a < b.
absolute value never equals
a negative, never less than a positive number (<)/ IxI=-2/ { }
axiom
a universally recognized principle
solving absolute value equations and inequalities
always 2 answers
counterexample
an example used to support a claim or statement that is the opposite of another claim or statement
When solving word problems for which an inequality must be used, look for phrases like the following:
at least, not less than > (:line underneath it) at most, not greater than < (:line underneath it) between, < between and inclusive, < (:line underneath it)
multiplying or dividing an inequality by a negative number changes the order of the inequality
changes the order of the inequality
In algebra, we write combined inequalities as one of two types of statements:
conjunctions and disjunctions
all real numbers in disjunctions "or" problems
covers all of the number line set notation: {all real numbers} interval notation: (negative infinity, positive infinity)
So, to solve an absolute value sentence graphically, Step 2
determine what distance "d" you need to move from "a" in each direction
So, to solve an absolute value sentence graphically, Step 1
determine what value "a" is subtracted from the variable and that number determines the middle of the graph.
equations (=) and "greater than" (>) inequalities are solved as
disjunctions, so write the 2 statements joined by "or"
absolute vale
distance from zero
Interval notation names the beginning and end of an interval
enclosed in brackets [inclusive] or parentheses (not inclusive). (It is useful to draw the graph first, then write the interval, reading left to right.)
irrational number has no
fraction
equivalent inequalities
inequalities that have the same set of solutions
Inequality that is not inclusive
interval: <, > graph: open circle O, ( )
Inequality that is inclusive
interval: <, > (with lines under them) graph: closed circle >, [ ]
any infinity signs
never inclusive, parentheses
try to mark the location of a
on the number line when solving absolute value sentences graphically
inequality
one variable is an open sentence formed by placing an inequality symbol ( ) between two expressions, at least one of which contains a variable.
a=
opposite of number on number line, negative to positive, positive to negative. Ex: Ix-3I= 2 a=3
So, to solve an absolute value sentence graphically, Step 3
place dots and/or shading on your graph from "a" to "a + d" and "a - d" depending on whether you have an equation "=", a disjunction inequality ">", or a conjunction inequality "<".
to solve an absolute value equation or inequality
remove the absolute value and replace it with =, get absolute value by itself
no overlap in conduction "and" problems
set notation: { } interval notation: no interval, O line through it
corollary
something that follows; a natural consequence
The following absolute value statements have specific geometric (graphic) interpretations because
the distance between two numbers on a number line is the absolute value of their difference
conclusion
the end or finish of an event or process.
U=
union (disjunctions, "or")
If an interval does not have a number as its beginning point
use (side ways 8) , negative infinity
If it does not have a number as its ending point
use (side ways 8) , positive infinity
To work with inequalities
use the Properties of Order (Properties of Inequality)(For all real numbers, a, b, c, d, (c 0):)
When we solve an inequality
we often write the solution sets as intervals on the real number line
Since the absolute value of a positive number is the same as the absolute value of its opposite
when we solve an equation or inequality that contains the absolute value of a variable, we have to consider both (two) possibilities
"less than" (<) inequalities are solved as conjunctions
write the 2 statements joined by "and"
inequality ex
x+2<4 or 2x+3>4x+7
absolute value greater than a number (>)
{all real number's, R"
a segment (<)
IpI< (line underneath it)5/ x=p, a=0, d=5/ solution set: {p: -5<(line underneath it) p<(line underneath it) 5}/ interval notation: [-5,5]
positive real numbers
(0, positive infinity (side ways eight))
negative real numbers
(negative infinity (side ways eight with negative in front of it), 0)
the real numbers
(negative infinity, positive infinity)
Empty set in set notation
-(10-r)<r-15/ answer: -10<-15 set notation: { }
examples of interval notation:
-2<x<5/ interval notation: (-2,5)
Example of solve and graph inequality 2
3(4z-1)>2(6z-5)/ answer: -3>-10 set notation: {all real numbers (capital R with 2 lines in it)} interval notation: (negative infinity, positive infinity )
Example of solve and graph inequality
3x+16<7/ answer:x<-3 set notation: {x: x<-3} interval notation: (negative infinity, -3)
Ix-aI=d
:in this statement x represents all the real numbers that are "d" units from "a"
Ix-aI>d
:in this statement x represents all the real numbers that are more than"d" units from "a"/ "or"
Ix-aI<d
:in this statement x represents all the real numbers that are within "d" units from "a"/ "and"
converse
If q, then p (if conclusion, then hypothesis)
Multiplication Property of Inequality:
If a > b and c > 0, then ac > bc. If a > b and c < 0, then ac < bc.
Addition Property of Inequality:
If a > b, then a + c > b + c.
To graph an inequality
place a subset of a line ( ray or segment ) on a number line.
conjunction
two or more statements joined by the word "and". To solve a conjunction, find the intersection (points in common) of the separate graphs. (different ways, smaller graph)
disjunction
two or more statements joined by the word "or". To solve a disjunction, find the union (all points on each graph) of the separate graphs. (same way, larger graph)