Math 051 lecture notes Professor Jason Samuels Addition and Subtraction of real numbers (1.3 & 1.4) ex) 3 + 5 = ex) 42 + 29 = ex) 12-4 = ex) 7-9 = ex) -3-4 = ex) 6 - (-2) =
ex) -5 - (-3) = ex) 7 + (-2) = you do: ex) 12 - (-7) = ex) -8 + 11 = ex) -9 + 4 = ex) 7-11 + 2 = ex) [(7-5) - (11-6)] =
Multiplication and Division (1.6 & 1.7) ex) 3 2 = ex) (4)(-3) = ex) (-3)(-6) = you do: ex) (6)(-5) = whats the operation? operation: multiplication sign - negative times negative is positive negative times positive is negative positive times negative is negative Compare: ex) 6-3 = ex) 6(-3) = you do: ex) -7-4 = ex) (-7)(-4) = some division: ex) 12 4 = ex) 21 (-7) = ex) 20 = -4 ex) -16 = -8
order of operations (or, do them all together) ex) 7 + 3 2 = ex) (4)(-5) + (-3)(-6) = ex) (3+6)4-8 you do: ex) 3-2(6-8) = parentheses: might mean - do me first - multiply - sometimes both Exponents ex) (2)(2)(2)(2) can also be written as: ex) 3 6 is the same as: ex) 5 0 = ex) 1 can also be written as: (6)(6)
opposite, reciprocal, absolute value (1.2) adding what number doesnt change what you have? 0 ("additive identity") ex) what do you add to -3 to get 0? thats called the additive inverse, or "opposite" multiplying by what number doesnt change what you have? 1 ("multiplicative identity") ex) what do you multiply by 5 in order to get 1? 1 / 5 thats called the multiplicative inverse, or "reciprocal" ex) whats the opposite of 7 / 5? whats the reciprocal? ex) what is 3? ex) -3 = ex) 2-5 = Properties of real numbers (1.5) commutative property ex) 12+25 = 25+12 ex) (5)(7) = (7)(5) associative property ex) (9+5)+6 = 9+(5+6) distributive property ex) 3(4+11) = 3(4) + 3(11) ex) -4(7-3) =
do it again... with fractions (1.9) Q. what is a fraction? -part of a whole -means of division -decimal -percent -rational number -one integer "over" another integer
why does adding fractions work like that? - here's the picture there's a '3' in the '6' so thats all you need
why does fraction multiplication work like that? - here's the picture
why do we use variables? - to represent (unknown) numbers - to help solve an equation - to show a general relationship ex) Sam is 18, Jason is 33 so, when Sam is 40, Jason is when Sam is x, Jason is two very DIFFERENT situations: expressions & equations we know expressions from arithmetic: ex) 4(7-5) what do you do? you calculate note that your final result is a number - feels like an answer there are expressions in algebra: ex) 3(2x-7) + 4 what do you do? you simplify. [why?] heres an equation: x + 5 = 8 note that your final result has x in it - not very satisfying how are EXPRESSIONS and EQUATIONS different? suppose you have 3x = 12 what do you do? you want to SOLVE FOR x you get x=4 you cant do that with expressions ex) 5x+20 what do you do? nothing (unless someone tells you to do something) 3x=12... what operation did we do? divide both sides by 3 can you do something like that with 5x+20? no: 5x+20 cannot magically become x+4 if you have an expression, there is no "=", and it cant magically appear
Evaluating expressions evaluate the following expressions ex) x 2 +4x if x=5 ex) 7 - x 3 + 3x if x=-2 ex) 2(4x-5)+3 if x=4 ex) 1 / 2 gt 2 if g=10, t=3 ex) 4(x-3) 2 if x=5 good general rule: use parentheses when you substitute (plug in) a value ex) x / 4 + 4x if x=-8 a word about exponents, parentheses, and signs ex) (-3)(-3)= ex) -3 3 = rewrite them with exponents: (-3) 2-3 2 so these are NOT THE SAME on the left, you square everything, including the negative on the right, you only square the 3 ex) (-2) 4 = ex) (-x) 4 =
lets talk about expressions (2.1) ex) 2x - 6 + 4 + 3x "pieces" are called terms they are separated by addition and subtraction compare: ex) 3 + x - 2 ex) 3x-2 what are "like terms"?...terms you can combine simplify: ex) 5+2x ex) 2x+3x ex) 3+2(4-3x) ex) 4x+3[2(9-3x)+4] now, ex) can you simplify x 3 + 3x - 2? are there any like terms? ex) 5m 2 + 3m - 2m 2 + m ex) 3x 2 + 2x 2 ex) 6t 2 + 4-3t 2 + t + 2
Solving linear equations (2.4, includes 2.2, 2.3) what does it mean to be "linear"?...if the expression only has x or y, nothing like x 2 or xy 2 (can be any variables, of course) solve for the variable: ex) x + 2 = 7 ex) x + 24 = 47 ex) t - 8 = 27 ex) x- (-4) = 15 ex) 4 = x - 19 ex) 2x = 6 ex) 12x = 48 ex) x = 15 3 ex) 7x = 84 ex) x = 12 5 7
more solving for the variable ex) 2x + 7 = 15 you want x by itself - what do you get rid of first? ex) 3x - 5 = 5x + 9 make it so that x appears on only one side of the equation ex) 4(x+2) = 2(3x+1) parentheses are in the way - so get rid of them... how? - by distributing you do: ex) -4x-3 = 13 ex) x+7 = 7x+31 ex) 2(4-x)+4 = 3(2x-4) now, with fractions ex) 2 x + 3 = 11 3 ex) 18 x = 720 100 you do: ex) 11 x = 55 20
solving for a variable, when there are two variables (2.5) if you are given the value for one variable ex) 2x - 5y = 30 solve for y if x=3 (thats just like the other problems) ex) 3x + 2y = 12 solve for x actually, this works exactly the same to get x by itself, what do you need to get rid of? ex) h + 4 = 5 solve for h 3 ex) x = 6 solve for x y ex) s = 4 solve for t t problem: t is in the denominator fix it: multiply by t on both sides ex) y+3 = 4 solve for y x
hw questions 2.2#19
Solving word problems (2.6, 2.7) you have to TRANSLATE between words and math ex) you have three. then you get two more. how many do you have now? ex) Dave has a certain amount. he gets seven more. how many does he have now? ex) Alan has a certain amount. it triples. how much does he have now? ex) Tanya has a certain amount. it doubles. then she gets four more. how much does she have now? ex) How much do Dave, Alan and Tanya have together? percents ex) 50% of 12 is what? ex) 15% of 80 is what? ex) 20% of what is 15? ex) at a restaurant, you got great service and want to give a 20% tip on a $60 bill. how much is the tip? ex) at the pharmacy you buy some aspirin. the sales tax on the purchase was $2. sales tax rate is 8%. how much was the aspirin sticker price? you can do the calculations using decimals or fractions by hand, i think fractions are easier by calculator, decimals are easier...its your choice
ex) a rectangle has a length 5 inches longer than it is wide. the perimeter is 58 inches. what are the dimensions of the rectangle? ex) Joel has twice as many nickels as dimes, all together worth a total of $1.80 how many does he have of each? first, give each unknown a name next, write down the relationship ex) two consecutive integers add to 45. what are the numbers?
inequalities (2.8, 2.9) ex) solve for x: 2x = 6 graph the solution on the number line: now, ex) solve for x: 2x > 6 solve it exactly the same way graph the solution on the number line: ex) 3x-7 < 1 everything is the same, with ONE EXCEPTION which is bigger? ex) 3 4 multiply both sides by (-2) ex) -6-8 so, when you multiply (or divide) by a negative number, the inequality switches note that this is the only time ex) 3<4, subtract 1 from both sides, you still have 2<3 ex) -3x < 15 compound inequalities [optional] how do we write "x is between 2 and 5"? thats the same as saying that its bigger than 2, and also less than 5 we could write: x>2 and x<5 heres a shorter way: take this 2 < x combine it with this x < 5 to get this 2 < x < 5 solve: ex) 2 < 3x-7 < 14 its easy, do them both at the same time
hw questions 2.6#17 jack is twice as old as lacey. in three years the sum of their ages will be 54. how old are they now? 2.6#37 tanner has $4.35 in nickels and quarters if he has 15 more nickels than quarters, how many of each does he have?
linear equations, representing and graphing (3.1-5) ex) cell phone monthly bill Shyanna's bill: $39.99 (and all the calls she wants) Tarik's bill: 10cents/min Jerry's bill: $29.99 plus 5cents/min which plan is best?...depends how much you talk what about 250min? Shyanna: $39.99 note: in terms of dollars "10 cents" is ".10" Tarik: (250)(.10) = $25 Jerry: (.05)(250) + 29.99 = 12.50 + 29.99 = $42.49 what about 550 min? Shyanna: $39.99 Tarik: (.10)(550) = $55 Jerry: (.05)(550) + 29.99 = 27.50 + 29.99 = $57.49 how can we represent this information? - numerically # minutes Shyanna Tarik Jerry 0 39.99 0 29.99 250 39.99 25 42.99 350 39.99 35 47.99 450 39.99 45 52.99 550 39.99 55 57.99 - graphically Shyanna Tarik Jerry
how much does it cost to talk 0 minutes for: Shyanna? Tarik? Jerry? what does that have to do with the graph?...where x=0 is where the graph crosses the y-axis - that is called the y-intercept Tarik pays.10 dollars per min, and he also pays $10 for 100 minutes is that the same thing? how can we check? in fact, we can do this with any two points and this tells us the rate, or slope ex) ex) note that this is different from calculating y / x...thats assumes that we started at (0,0), which may not be true [economics sometimes uses this calculation] for Shyanna's plan, what is her rate? is it always the same? for Tarik's plan, what is his rate? is it always the same? for Jerry's plan, what is his rate? is it always the same? a linear function has the same rate (or slope) everywhere what is the function for each plan? Shyanna: Tarik: Jerry: in general, for a linear function, we can write y = mx + b m = slope b = y-intercept
examples of linear functions ex) y = 3x+2 graph: slope= y-intercept= some solutions: x y ex) y = 2x-1 solutions can be written as table or ordered pair ex) y = -2x+3 if slope is negative, @ line is decreasing (going down) @ the graph will go "down & right" (not down & left) if you have two points, call them (x 1 y 1 ) and (x 2 y 2 ) slope = change in y = y 2 - y 1 change in x x 2 - x 1 also called rise/run
also equations of a line: ex) 2x + 3y = 12 graph: slope: y-intercept: solutions: ex) x - 3y = 6 graph: slope: y-intercept: solutions:
finding the equation of a line (3.6) find the equation of a line given the slope and y-intercept ex) slope=3, y-intercept=4 (pretty easy) find the equation of a line given the slope and any point ex) slope=2, point is (3,4) method 1: use y=mx+b we know the slope, so we have y = 2x+b we can plug in the values from the point and it will satisfy the equation 4 = 2(3) + b 4 = 6 + b -2 = b so the equation is: y = 2x - 2 method 2: the answer is y - 4 = 2(x-3) first, where does that come from? second. how can there be two different answers? second question first: its really the same formula: y-4 = 2(x-3) y-4 = 2x-6 y = 2x-2...aha! now, where did that come from?...it comes from the slope (here's the explanation, if you are curious) we know that the slope is 2 2 = y 2 - y 1 x 2 - x 1 but we have values for one point, so (x 1 y 1 ) = (3,4) 2 = y 2-4 x 2-3 also, we want this to be true for any point on the line, that is (x,y) 2 = y - 4 x - 3 cross-multiply to get y - 4 = 2(x-3) or, just remember the formula: y - y 1 = m(x - x 1 ) ex) find the equation of the line with slope=-4, through the point (5,2)
find the equation of the line using two points ex) find the equation of the line through (2,7) and (4,13) we can find the slope: now we have the slope and a point (pick either one), so its like the problem we just solved you do: ex) find the equation of the line through (-2,3) and (1,9) special cases -horizontal lines graph: whats the slope? whats the equation? -vertical lines graph: whats the slope? whats the equation? note that the slope is undefined - you cannot write the equation of this line as y=mx+b
hw questions ch3
Solving TWO linear equations with TWO variables three ways: by graphing (4.1), by elimination (4.2), by substitution (4.3) here's what a problem might look like: ex) suppose you sell concessions at a cinema. you sell small sodas for $2 and large for $3. you ran out of small cups, so you only have one size cup today (for a small order, you fill it halfway). at the end of the day, you sold 86 sodas for $191. your manager wants to know how many of each size you sold. can you tell him? what do we know? the situation: - two variables - two equations...what do you do? simplify the situation so you have one variable and one equation (...once you have that, you know what to do) was that magic? no, lets learn how to do it.
ex) x = 6 solve for x & y y = 2x+3 ex) y = x-2 y = 2x-6 solve with algebra: (thats too easy) solve with a graph: the solution tells us the values for x,y that work in both equations ex) y = 4x-3 solve for x and y y = x+9 ex) 2x+y = 7 solve for x and y y = 4x-11 how do we solve this?
ex) x - 2y = -7 3x + 2y = 3 do it again, a quicker way ex) x - 2y = -7 3x + 2y = 3 thats too easy - are you allowed to do that? consider: a = 3 b = 2 a+b =? c - d = 7 d = 2 c =? you do: ex) 2x - 3y = 5 x + 3y = 7 note: when you solve for the second variable, you need to plug a value into an equation. you have two choices, it doesnt matter which equation you use
now, ex) 2x - y = 9 x + 2y = 2 what do you need to have so that one variable gets eliminated? ex) 2x + 5y = 9 4x + 5y = 9 heres the key step: to eliminate x, we want the 'x' in the second equation to have a coefficient of '-2' so, multiply by -2 whatever you do to one side of the equation, you do to the other, so its legal you do: ex) 2x + 3y = 5 3x + y = 11 note: some tricky person might put "y" first......make sure x's are lined up and y's are lined up when is it easier to you use... substitution?...when one variable is already by itself elimination?...when one variable is set up to cancel (the coefficients are same number, opposite sign) ex) 2x - 4y = -10 solve for x & y 3x + 2y = 1
special cases: ex) x+2y=3 x+2y=4 ex) 2x-3y=4 4x-6y=8 ~~~ same thing, with word problems (4.4) ex) John has $1.70 in dimes and nickels with 22 total coins. how many of each does he have? ex) at a cinema, adult tickets are $12, child tickets are $7. if 110 tickets are sold for $1150, how many adults and children came?
ex) the difference between two numbers is 12. their sum is 38. what are the numbers? ex) one number is one more than triple another number. their sum is 29. what are the numbers? ex) two consecutive integers add to 45. what are the numbers? ex) two consecutive even integers add to 74. what are the numbers?
hw questions 4.1#1x+y=3 solve by graphing x-y=1
when there are ( ), raise everything in the ( ) to the exponent when there are no ( ), only raise the number (or variable) without the sign
combining for addition: same base, same exponent combining for multiplication: same base
not -9 once you 'flip it' you have taken care of the negative
"8 subtract 5"... 8-5 "8 subtract -5"... 8 + 5 if you "stack it up" and you are doing subtraction, remember to flip the signs
ex) start with 3x 4 +4x 2-7x+4 and subtract -4x 4 +6x 3 +3x 2-4x+2 ex) simplify: (x 2-3x + 2) - (3x 2 + 5x - 3) ex) subtract x 2-4x+7 from 3x 2-2x-6 note that "subtract A from B" means B - A watch out!
5.6 special cases ex) (x+4) 2 ex) (s-5) 2 ex) (3x+2) 2 ex) (4x - 7y) 2
5.7 dividing a polynomial by one term ex) 3x 4 + 2x 3-7x 2 x 2 ex) 4x 5-10x 3 + 8x 2x you do: ex) 12x 5 + 9x 4 + 15x 3 3x 2 ex) 10x 4 + 20x 3-15x 2 5x 2 ex) 16x 4-20x 3 + 16x 2 4x
hw questions ch5 5.7#72
review for midterm
ch6 - Factoring opposite of multipliying out recall: ex) multiply 3x(x+4) answer: 3x 2 + 12x ex) multiply (x+5)(x-7) answer: x 2-2x - 35 now: ex) given 3x 2 + 12x, get 3x(x+4) ex) given x 2-2x - 35, get (x+5)(x-7) how do we do that? 6.1 factoring - find the gcf factor: ex) x 2 + 3x ex) 3x 4 + 7x 3-4x 2 ex) 4p 2 + 12p + 20 ex) 6r 3-18r 2 + 6r ex) 6a 2 b 3 + 2ab 4 + 2a 4 b 2 look at: coefficiencts "a" exponents "b" exponents you do: factor ex) 8x 3-16x 5 + 20x 2 ex) 12r 3 s 4-6rs 5
6.2 some basic factoring x 2 + 5x + 6 = (?? ) (?? ) how do we do this? first we make an observation: the lead term is x 2... (what) times (what) will give x 2 x 2 + 5x + 6 = (x )(x ) how do you do this?...first, look at last coefficient, figure what two numbers multiply to that (many possiblities) next, look at middle coefficient, find which pair add to that ex) x 2 + 6x + 8 ex) x 2 - x - 12 ex) x 2 + 2x - 8 ex) x 2-49 ex) x 2 + 10x + 25 note: sometimes you CANNOT factor ex) x 2 + 2x + 8 try to factor it: ex) x 2 + 4x + 7...and when i say "CANNOT", i mean you dont know how. you can factor this with advanced techniques which are beyond the scope of this course
ex) 6r 3-18r 2 + 12r you do... factor: ex) 2x 2 + 18x + 40 ex) 4x 2 + 4x - 24 HOW TO FACTOR (so far) 1. find gcf (greatest common factor) 2. x 2 "x and x" 3. product: break last term down into factor pairs 4. sum: look for the pair which adds/subtracts to the middle term 5. check
6.3 factoring - when the lead coefficient is NOT 1...a little harder ex) 6x 2 + 19x + 10 1. always check for a gcf first [here there is none] 2. multiply first and last coefficient 3. factor that into number pairs, look for which ones add to the middle number 4. rewrite the original polynomial by breaking up the middle term 5. for the first two terms and the last two terms, separately factor the gcf 6. factor again (if you did it right, the same factor will appear twice) lets check the answer:
ex) factor 4x 2 + 16x + 15 how will question be asked on an exam? on the final: "factor completely" on the Compass: "which of the following is a factor of 4x 2 + 16x + 15?
you do ex) factor 10x 2 - x - 3 dont forget the +1 check your work by multiplying factoring - with two variables how do you factor 6x 2 + xy - 2y 2...do it as though there is one variable
ex) factor 4x 6 + 12x 3 y + 5y 2 recall: ex) factor x 2-81 now, ex) factor 4x 2-25 ex) factor 9r 2-16 ex) factor 64x 2-9y 2
hw questions 6.2-30) factor 2a 2-18a + 28
using factoring to solve equations ex) solve for x: (x-2)(x+3) = 0 what happens if you try to get x by itself? we need a new method... solve for x: (x-2)(x+3) = 0 something times something equals zero...so one of them must be zero - aha! x-2=0 or x+3=0 x=2 or x=-3 you get two solutions! how can we use that in other sorts of problems? ex) solve for x: x 2 + x - 6 = 0 moral of the story: suppose you have an equation where you must solve for x if the equation is NOT LINEAR (like if there is x 2 ), you cannot get x by itself the way to solve this is to FACTOR
solve for x: ex) x 2 - x - 12 = 0 ex) 2x 2 - x - 10 = 0 ex) x 2 + 2x - 7 = 1 ex) x 3-5x 2 + 4x = 0
a review of factoring: ex) factor x 2 + 2x - 24 ex) factor 6x 2 + 11x - 10 solve for the variable: ex) s 2 + 2s = -1 ex) 8x 3-18x 2-18x = 0 notice that this problem: 2x + 3 = 5x is different from: 2x 2 + 3 = 5x...to do the second one you need to factor (that little 2 makes a big difference)
hw questions - ch6
ch7 rational expressions 7.1 reducing we call this "canceling" but actually we are dividing terms: things we are adding factors: things we are multiplying
reduce: cant "cancel" x...why? cant cancel 2 and 6... why? reduce: reduce: factoring gives you something times something, so you can divide and cancel
see what happens when you multiply by (-1) OR factor out (-1) directly if there are *signs* and you are not sure, multiply one factor by (-1) and see if it gives you the other factor COMPASS TIP: check your answer by plugging in a number
check by plugging in a number:
7.2 rational expressions: multiply and divide
first step: combine fractions...do you need LCD? second step: simplify Same denominator - just add the numerators
7.4 rational expressions: solve equations
hw questions ch7
homework questions ch7
ch8 roots and radicals what is a radical?...the opposite of a power
roots and signs
combine and simplify radicals 8.5 multiply, divide 8.3 simplify 8.4 add, subtract
for multiplication, you do not need like terms to combine for addition, you do same idea here, its just that the terms are more complicated (because of the radicals)
hw questions ch8
7.6 complex fractions how do we simplify this?...well, "fraction" is the same as "division", so rewrite it as division - then solve it as you normally would
7.7 proportions and rates ex) if a car travels 120 miles in 2 hours, how far would it go in 5 hours (at the same rate)? ex) if Sam can read 240 pages in 3 nights, how much can she read in 8 nights (at the same rate)? ex) on a map, 1 inch represents 85 miles. 3.5 inches represents how many miles? ex) a baseball player gets 15 hits in 20 games. at that rate, how many hits will he have in 84 games? after how many games will he have 78 hits?
distance what is the distance between two points?...to find it we use the pythagorean formula ex) what is the distance between (2,4) and (6,9)? c 2 = a 2 + b 2 to make this a formula for just the distance, we write: c = a 2 - b 2 what is a? what is b? so, the distance is c = lets make that one formula: for two points (x1 y1) and (x2 y2), the distance between them is c = (x2 - x1) 2 +(y2 - y1) 2 ex) find the distance between (5,2) and (4,6) ex) find the distance between (4, -1) and (-3,2) check your signs!
hw questions 7.6, 7.7
review for final