algebra comprehensive

what type of brackets are used for which inequality symbol <> ≤≥

<>( ) ≤≥[ ]

√x

⌒

slope is the _____ of change between two points

rate

a set of ordered pairs is called a ....

relation

(a + b)² = (a − b)² =

(a+b)(a+b) (a-b)(a+b)

how do you solve these types of problems: (list the steps) solve and graph: |3x|>-6

-set up two inequalities where the non coefficient number (-6) is a positive and in the other equation it is a negative -SWITCH THE SIGN WHEN YOU MAKE IT THE OPPOSITE OF THE ORIGINAL 3x>-6 ; 3x<6 isolate x x>-2 ; x<2 graph and write it in interval notation (-∞,∞)

a⁰=

1 Anything raised to the 0 power equals 1

general form: a linear equation in two variables written in what form

Ax+By=C

slope = (equations and r+r)

rise over run y2-y2/ x2-x1

parallel lines have the _____ slope and ____ y-intercepts perpendicular lines have ____ and _____ slopes

same + different opposite reciprocal

x²

U

|x|

V

if it is a true statement with inequalities, x goes away but it is still true (ex 6>1) it is

all real numbers

at least at most between no more than no less than

at least (x≥5) at most (x≤5) between (5<x<10) no more than (x<5) no less than (x>5)

(a-b)(a+b)

a²+b²

a²+a³=

a²⁺³ or a⁵

(aⁿ)²= (ab)ⁿ= (a/b)ⁿ=

aⁿ*² aⁿ+bⁿ aⁿ/bⁿ

two inequalities joined by and or or are called

compound inequalities

Three planes intersect at all the points of a single line. This system has infinitely many solutions and is said to be

consistent

When 3 planes have a single point in common. This point is the solution to the system and we say the system is

consistent

Distance Formula

d= rate x time

Three planes coincide at all points on the plane. The system is consistent, and the equations are

dependent

compound inequalities, when is your answer in set builders notation { } and when is your answer in interval notation [ )

set builders notation { } is when you are finding the U or ∩ between two sets (ex: a={x|x is an even number} b={1,2,3,4}) interval notation [ ) when you find the the U or ∩ between inequalities (ex: 2x≥0 and 4x-1≤-9)

The absolute value of any expression is always ____ or ____ to zero

greater than , equal

2. Three planes intersect at no point in common to all three. This system has no solution and is said to be

inconsistent

when solving systems of equations: the resulting equation is false then we have an ______ system and the solution set is {} or ∅

inconsistent

when you divide or multiply by a negative number what has to happen

inequality symbol must flip

when solving systems of equations: if the resulting equations is true (ex: 0=0; 5=5) we have _____ solutions and the solution set is {(x,y)-5x-3y=4} or {(x,y) insert one of the equations}

infinite

intersection and union

intersection: ∩ elements common to both sets union: U belong to either or of the sets

∈

is an element of

what's the first step of this problem |2x|+5=7

isolate absolute value/ subtract 5

degree is the

largest exponent

when solving inequalities: this means < and this means > when do you decide what the inequality is asking w/ absolute value inequalities

less than AND greater OR AFTER YOU ABSOLUTE VALUE IS ISOLATED

how to solve: find from the graph to the right g(-2)

look at the graph find -2 on the X AXIS and find where it is on the line where -2 on the x axis meet and whatever number on the y axis your answer will be (-2, corresponding y number)

when trying to find the price before tax

look at the tax % and add that too 100 because you are paying 100+ the tax ex: tax= 7% (0.07) you divide the orginal price by 1.07 or (107%) because you are paying the full price (100) and the tax (7%)

How to solve this: Your younger sibling has a coin collection that consists of 14 coins with a total value of $1.35. If the coins are nickels, dimes, and quarters, and the number of nickels is 3 less than twice the number of dimes, how much of each coin is there in the collection?

n+q+d=14 n=2d-3 0.05n+0.10d+.25q=1.35

if a is a negative number, then |x|=-a is

no solution

where the two lines intersect

origin

opposite of a number is

positive to neg

functions: identifying if the relation is a function Numbers in ___ can repeat, but number in ____ cannot repeat if NO _____ can be drawn so that it intersects a graph more than once then the graph is a function

range, domain vertical line

How to solve this: Two cars leave Indianapolis, one traveling east and the other west. After 3 hours, they are 297 miles apart. If one car is traveling 5 mph faster than the other, what is the speed of each?

set up two equations: x=y+5 3x+3y=297 set the equal solve

if it is an expression ____ if it is an equations _______

simplify, solve

It's best to use... Slope Intercept Form: When you are given ____and ___or _____ and _______ Point Slope Form: When you are given ____________ and a ____________ or __________ _____________.

slope , y intercept or two points and y intercept point slope: slope point or two points

graphing linear equations: creating a table pick any multiple number for x and ....

solve for y w/ your x's, choose neg and positive number for x

when trying to the % off you would

subtract the percent from 100 and divide (finding the starting point) or multiply (finding how much you pay) it to the original price

counting quadrants: where is number one, which direction do you go

top right proceed left then down then right make a C shape 21 34

domain range

x y

3 consecutive integers 3 consecutive odd integers

x+(x+1)+(x+2) x+ (x+2)+(x+4)

piecewise functions and shifts: x+k = x-k= (x-h) (x+h)

x+k = move up k x-k= move down k (x-h)= move right h (x+h)= move left h

How to solve this: mixing solutions How much 20% solution and 50% solution must be mixed to get 12 gallons of a 30% alcohol solution?

x+y=12 .2x+.5y=3.6 solve

if a is a positive number, then |x|=a is equivalent to

x=-a x=a

are these functions x=2 y=-3

x=2 no y=-3 yes

point slope form + how to use

y-y1=m(x-x1) if you are given two relations find the slope between to two using the slope equations (y1-y2/x1-x2) then plug that into m use ONLY ONE of the two relations and plug that is for y1 and x1 then solve and end in the form y=mx+b

to find an x-intercept let ____ and solve for x to find a y-intercept let ___ and solve for y

y=0 x=0

slope intercept form what does each variable mean

y=mx+b m= slope b= y intercept

solve |y|=0

{0}