1.1

• Numbers

  • Rational Numbers

    • ½ , -3/7, 46, 0.77 - shown as fractions

  • Integers

    • … -3, -2, -1, 0

      • Natural Numbers

        • 1, 2, 3

  • Irrational Numbers

    • root3, root5, pi - non repeating fractions

• Properties

  • Communitive Prop of Addition

    • When we add two numbers, order doesn’t matter.

    • inverted

  • Communitive Prop of Multiplication

    • When we multiply two numbers, order doesn’t matter.

    • inverted

  • Associative Prop of Addition

    • When we add three numbers, it doesn’t matter which two we add first

    • same order diff ()

  • Associative Prop of Multiplication

    • When we multiply three numbers, it doesn’t matter which two we multiply first.

    • same order diff ()

  • Distributive Prop

    • When we multiply a number by the sum of two numbers, we get the same result as we get if we multiply the number by each of the terms and then add the results.

• Notations

  • set builder notation

    • A = { x | … }

  • interval notation

    • ( denotes and open

    • [ denotes a closed

    • such as but not limited to, (0,0), [0,0], (0,0]

• Inequalities

  • the real number lines

  • numbers that are less are on the left

  • numbers that are more on the right

  • a < b when a is less than b

  • b > a when b is greater than a

• Sets

  • U - union

    • all numbers in a given set

  • upside down U - intersection

    • numbers that are the same - “intersect” - in a set

  • be careful when writing them out - make sure graph is correct

• Intervals in terms of Inequalities

  • ( open also OPEN circle

    • < or >

  • [ closed also CLOSED circle

    • <= or =>

• Absolute Value

  • |a| and a is positive = +

  • |-a| and a negative = -(-a) = +

  • used to show distance

• Hardest: 77, 86, 87

1.2

• Radicals and Exponents

  • Exponents

    • power of 0 = 1

    • power of 1 = base number

    • multiplying exs = adding

    • dividing exs = subtracting

    • () = multiplying

    • negative = move to the denominator or vice versa

  • Radicals

    • ¼ = 4root

    • 2/3 = 3rootx²

    • radical x radical breaks free

    • get radical out of denominator by multiplying by root top and bottom

  • Scientific notation

    • move decimal —→ negative

    • move decimal ←— positive

  • hardest: 68, 75,

1.3

• Factoring

• refer to notes if needing special equations

• pretty simple/self explanatory

• good at this <3

• GO OVER SPECIAL RULES

• hardest: 33,44, 79, 136

1.4

• identifying domain

  • anything in the numerator is = all real numbers

  • anything in the numerator + root = >=0

  • anything in the denominator /= 0

  • anhting in the denominator + root >0

• simplifying rational expressions

  • numerator/denominator (could be multiplied by another)

    • factor and cancel common terms

  • if dividing two —> flip the second and multiply + factor + cancel

• adding and subtracting two fractions

  • find LCD

  • multiply numerators by the denominator factors they don’t already have

  • simplify

  • factor and cancel

  • simplify all the way

• compound fractions

  • same steps as before just x2

    • take the bottom fraction flip and multiply by the first

    • factor and cancel

    • simplify all the way

• rationalize the denominator

  • if just root —> multiply by root to break free on top and bottom

  • if cube root and x² —> multiply by cube root x (x² x x break free) on top and bottom

  • if two terms (one real and one root) multiply by the pos (if neg) or by the neg (if pos)

  • same for numerator

• hardest: 90,79

1.5

• linear equations

  • plug in x when given and solve

  • fractions with linear equations

    • multiply numerators with both denominators to cancel out

    • simplify —> factor —> cancle

      • simplify further

• solving for variables

  • get the variable alone

• solving using factoring

  • factor then set x to 0 and solve

• completing the square

  • move the real number to the zero side

  • add (b/2)² to both sides

  • simplify the perfect square and solve

  • make sure root is PLUS AND MINUS

• quadratic equations

  • -b +- root b² - 4ac/ 2a

• using the discriminate to determine how many answers

  • b² - 4ac

    • - = none

    • + = two

    • 0 = one

• absolute value

  • set the = to neg and pos

    • solve —> two x’s

• CHECK SOLUTIONS

• hardest: 118, 133

1.6

• sum and difference of imaginary numbers

  • in the form a+bi

• mulitply the two with the FOIL method

  • simplify i

• single fraction with in in the denominator

  • do what you would do when rationalizing the denominator

  • simplify i

• similar to complex fractions

  • simplify i

• cannot multiply two negative roots

• can use the quadratic equation

• conjugates

  • change the sign connecting the two terms

• hardest: 51, 55,

1.7

• word problems.

• d = rt

• hardest: 70 and 64 hw and last two on notes

1.8

• Linear inequalities

  • get x alone

  • graphing:

    • < or > means open circle

    • <= or => means closed circle

  • set notation

    • < or > means ()

      • infinity is always ()

    • <= or >= means []

• nonlinear inequalities

  • kind of like factoring where you get two x’s

  • make everything on one side and 0 on the other

    • < or <= determines the sign of the inequality answer

      • open circle/closed circle and ()/[]

  • check each portion of your graph by inputting a number either higher/lower/in between to determine where to draw ur line

  • the set notation depends on the last step

    • if lines go ←— and —→ without connecting in between you will have two set notations with U in between

  • DO THE SAME IF YOU HAVE 3 X’s

    • check all

    • if one point doesn’t have a line do {number} with U in between

• nonlinear inequalities with fractions

  • when everything is on the correct side

    • set x = 0

      • once you get your factors you have to make sure they are included in the inequality answer (so open or closed circle)

        • if it is in the denominator (the factor) it has to be an open circle

        • RULE ONLY APPLIES IF OG INEQUALITY USES <= or =>

          • if of uses < or > there is no need to check

    • same rules as before

  • when inequality is not set to 0

    • move it to other side and multiply numerator and denominator by og denominator so you have one fraction

    • then follow same steps as before

  • if there is an x² that is an additional root

• absolute value inequality

  • works the same as a regular

  • you’ll get your immediate answer after solving

    • graphing is easy

• hardest: 87,101,

1.9

• graphing domain and range

  • y is horizontal

  • x is vertical

  • included in graph = filled in line

  • not included in graph = dashed line

• distance

  • d = root (x2-x1)² + (y2-y1)²

• midpoint

  • (x2+x1/2, y2+y1/2)

• Areas

  • rectangle/square = lw

  • trapezoid = (b1+b2/2)h

• isoceles

  • two sides of the triangle are the same

• given midpoint and not the second point

  • set midpoint x = (x2+x1/2)

    • input what you have and solve for x2

    • same for y

• y and x-intercepts

  • set x = 0 for y-intercept and solve

  • set y = 0 for x-intercept and solve

• testing symmetry

  • x-axis

    • set y to -y

  • y-axis

    • set x to -x

  • origin

    • set x to -x and y to -y

  • if they equal the of equation it has that symmetry u tested for

• when using square root for x and y intercepts

  • if both sides need to be squared / rooted ex: x² = 9 is when there will be BOTH negative and positive answer

  • if only one side needs to be rooted ex: x = root9 there will only be ONE positive answer

• equation of the circle

  • (x - h)² + (y - k)² = r²

    • the center is (h, k) —> positive if inside is neg

• graphing circle

  • find center and radius

  • graph the center, the circle will not extend past the radius on all sides

• hardest: 100,

1.10

• slope

  • y2-y1/x2-x1

• point slope form

  • y-y1 = m(x-x1)

    • (x1, y1) point on the graph

    • m is slope

• slope intercept form

  • y = mx +b

    • m is slope

    • b os y-intercept

• finding an equation perpendicualr to the line

  • find given slope —> find the opposite

    • if positive —> negative

    • if a fraction/number —> flip it

      • must do both of those

  • put given points and new slope into point slope form

  • simplify to slope intercept form

• finding an equation parallel to the line

  • find given slope —> slope stays the same for new equation

  • put given points and same slope into point slope form

  • simplify to slope intercept form

• if given an equation like x = 5

  • slope = undefined

  • y-intercept: none

  • graph is a verticle line

1.12

• directly proportional

  • y=kx

  • y is directly proportional to x

• inversely proportional

  • y=k/x

  • y is inversely proportional to x

• input any given values

  • answer should be the variable and k as a number

• if it says the two variables = the same just make them the same variable to solve

  • the actual equation will have the actual variable

• when asked how y is changed when multiplying the other variables

  • ignore the other variables and put in the new values

    • multiply and simplify

    • take the real numbers out of the equation

      • answer

• hardest: 35

PRACTICE TEST HARDEST: 7, 11, bonus