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)