Complete Algebra Mock Final Review: Key Problems and Practice Strategies

hello everybody Mr fronte here and here's the video to go over the entire mock final and give you more of a chance

to practice so we're going to check answers so you know which problems you got right and which things you need more

practice on I'm going to do the problems myself and explain in more detail each of those problems and finally as we

learn we're going to go ahead and give you a chance to practice more or do further learning on YouTube so first

let's check answers you can pause the video at any time so here are just quick answer

checks so number one two through eight is right there pause the video as needed then here's the rest of them 9 10 11 12

13 14 and there's number 15 so pause the video double check your answers and let's now do each of these problems in

particular and then show you how to practice more on your own so I'm going to go ahead and split my screen here

view split screen and down below I'll show you in a bit here I'm going to basically have a reference

for what to search in YouTube to get more practice so if I scroll down to my document um I'll explain this in a

second all right let's start by doing the first problem now for these kinds of problems the first thing you need to do

is just know the different expressions and translations of the individual words so in this case um there's two by itself

translates to the number two okay less than less than in this case means something minus if you were

thinking that's this less than this is for something is less than so is less than is an inequality but if you just

say less than by itself it means subtract for example if someone makes $30 per hour and their coworker makes $5

less than them that would be $25 right so in everyday English the word less than means to subtract and

that's what it is here so let me erase my writings here and let's continue translating the

individual pieces of the sentence so two is the number two less than already means subtract and so already I can

start writing an expression if it's two 2 less than something it's already something minus 2 that is two less than

something what is two less than is this here I'll put it in blue two less than the product of X and Y so again each

word in here translates product means multiply and in this case it's X and Y that you're multiplying together so we

have two less than the product of x * y I'll write it as XY and then the word is I'll put this in green is translates in

this case to equals so all this stuff is equals the quotient of x and y so quotient means

the answer to a division problem and the two things you're dividing in this case are Y and five in the same order so is

is equals the quotient division of two things in this case Y and 5 so that's the translation of this sentence and you

really have to break down each word into like what symbol now to practice this more if I scroll down my document it's

why I split screens you can see here to practice more type in either of these two search phrases into YouTube to get

tons more problems and here's an example so I'm going to type in this search phrase into YouTube

and then what you're going to get is a lot more videos that you can either learn for from or practice so here's a

great example I'm going to click on the first video here so if you feel like you want to pra

to learn more then you can just watch the video outright and don't worry about um trying the problems

yourself so all right so if you just want to practice but not learn like I watch the

teacher here's what to do do the problem yourself so pause the video try the problem

yourself and then play the video that way you'll get a comparison of like what you thought versus what the teacher

thought so and then you can skip ahead in the video to the next problem once you've done this

one so here's another one right six less than five times the number is nine what's the number you would translate

this yourself posi video translate it yourself and then just play the video or skip forward to see if your translation

was correct that way you can practice a ton of problems on YouTube and if you mess something up the video will explain

each step for you so you can see where you went wrong this is a great technique that works all the time for pretty much

every topic in in our class so I encourage you to use this as much as possible to to learn algebra all right

number two evaluate this expression when X is 7 and Y is 2 so these kinds of problems come down to just

substitute numbers for x and y and follow the order of operations so as a reminder we're going to need to follow

PEMDAS the order of operations so pemos stands for parentheses exponents multiplication and division and then

addition and subtraction so let's first substitute 7 for x and 2 for y we've got three times whenever I substitute I use

parentheses so X is s so I'm going to put that in parenthesis minus the Y value is

two and it's that whole thing squared plus X is 7 and Y is 2 so I'm done substituting 7 for X and Y for 2 two for

y and next step is to follow pemos so first are there any parentheses we have to handle first yes there are before we

can do exponents like this one here the power of two before we can multiply we need to take care of the parentheses so

inside this big parentheses there's a 7 minus 2 so 7 - 2 needs to be done before

multiplying the three and before doing the power of two because that's an exponent and for I'm just going to keep

these the same that's 7 * 2 all right 7 - 2 is 5 so we got 3 * 5^ 2 7 time two there all right now that

we're done handling the information inside parentheses now we can move on to exponents and this is the first place I

see students have a question they're like well should I multiply by the three or should I do the power first the

answer is right here you have to do exponents first first before multiplying so let's do the exponent first that's

squaring 5^ squar means 5 * 5 and 5 * 5 is 25 and then finally these all stay the

same and now we're down to Let's multiply everything that uh has to be multiplied 3 * 25 is

75 and then 7 * 2 is 14 and then finally there's no division but there is addition so we can move on to addition

75 and 14 is uh 89 so that is our final answer and to practice more go ahead to YouTube and just type in either of these

two search phrases and then when you see a problem pause that video try the problem yourself and then resume the

video and see if you correct all right next up number three so solving for

x this would be the usual steps where it be wise okay first of all let's just do this let's do this let's make a

mistake on purpose if a student says hey we should subtract six first that seems at first that seems completely

reasonable but in reality this actually isn't six it's two times all this stuff actually it's 12 because it's twice X+ 6

so this plus 6 is deceptive it's not really six it's actually 12 so the first step you need to do when you're solving

for x is make sure you distribute any parentheses so distribute to any parentheses in our case 2 * X is 2x 2 *

6 is 12 still Min - 9 and still equals xus one I'm going to draw a line separating the two sides of the equation

once you've distributed you can combine terms 12 - 9 is 3 and it's still 2x plus that 3

and then xus one and now our goal is to get X on the same side and to get X to be alone so you can either subtract 2x

to both sides that'll be a legal step if you do this you're going to get --1x here minus

one I'm going to choose a different step this step Works it'll work if you want to try it go for it I'm going to back up

though I'm going to choose a different step I'm going to choose to subtract One X from both sides and the reason I'm

choosing this is because 2x - x is just 1X that's going to be easier at the end because there's no

negative with it so that's my choice all right final step subtract three to both sides those cancel out and I've got x =

-4 done so to practice more problems in YouTube Type in either of these search phrases and there's going

to be tons of problems so for example here here I type in solve for x multiple steps in YouTube You're Going to get a

whole bunch of practice problems just like this and so that was fractions I would skip that for now we'll do it

later but there's a problem this one's almost exactly like ours right here so here's a great one if we play this video

I'm sure it'll be yeah there's ads so once the ad gets through here so let's scroll ahead to the first

problem perfect all right so you would pause the video try this yourself and then resume the video or fast forward it

until you can see if you were correct but it's a great way to check your answer and if you mess up anywhere you

can see in the video which Step you made the error on so it's a good way to practice all right moving

on okay number four so number four you can solve two ways you can solve with a regular

solving for x or you can use fraction Busters and I'm going to do it both ways so fraction

Busters so here's both ways to solve this all right with regular equation solving you just do your regular steps

there's nothing different to do so first we could subtract three like usually you save all the X stuff for the end right

like over here we didn't deal with dividing or multiplying till the very end end in fact on the very end we

didn't even need to multiply or divide if this had been 2x = 4 we would divide by -2 but that wasn't necessary for that

problem but you usually save all the dividing for the end so that's what we're going to do here subtract three

first those cancel then we've gotx over 2 1 - 3 is

-2 and now we need to get rid of this negative like a half in front of X so I'm going to do it in just one step and

multiply by -2 to both sides that's because this -2 and that -2 ones in the numerator ones

in the denominator they cancel out leaving just X by itself on the right hand side -2 * -2 is pos4 and we're done

so that's just regular solving no fraction Busters okay fraction Busters is the technique of eliminating

fractions as your first step and so in this case the only fraction is a half so you want a number that two will go into

so you can multiply both sides by four by six by two any of those choices would work I'm going to choose to multiply

this equation all by two so two times this guy two times that guy and then two times that guy so I multiplied

everything by two but you could also multiply by four or by six and any number that two goes into will work so

for me I chose two all right 2 * 3 is 6 here the twos cancel out that was the whole point so it's just minus X is left

over 1 * 2 is 2 final steps subtract six first those cancel out you havex = 2 - 6 is -4 and then here we still need to get

X alone it's not alone yet Sox is the same as saying1 X so I will divide 1 to both sides to

cancel out that1 and you have left over x equals a negative divid negative is positive in

this case positive4 so you got the same answer either way to practice more fraction Busters then go ahead and

search YouTube for any of these key phrases and you're going to get tons of videos to pause the video try those

problems yourself and then resume the video to see if you are correct all right number five and then

six number five is definitely intended for fraction Busters so for number five we're going to use fraction Busters what

number can nine and three both divide into so nine goes into something and so is three um 27 that would totally work

because nine goes into 27 and so is three so that a possibility there a smaller possibility so even nine would

work because nine goes into itself and so does three so the easiest possibility is nine but if you chose 27 or a bigger

number that nine and three go into it would still work but I'm going to choose nine because it gives you the easiest

numbers to deal with all right so multiplying everything by nine 9 times 9 times everywhere this

is 9 over one the nines cancel out leaving two 2 x uh 3 goes into 9 3 * 3 * 1 is 3

remember it's still minus so minus three because it was a minus before so it's still minus and 2 * 9 is 18 all right

final steps let's add and subtract before dividing so plus three to both sides would be the easiest First Step

those cancel out 2x = 21 divide both sides by two and x equal 21 Hales and you can just leave it as 21

halves because it doesn't reduce any farther all right that is correct the answer in this case is a fraction and to

try more problems with fraction Busters go ahead and search YouTube for any of these expressions and you'll get more

problems just like this one to practice all right number six this is still fraction Busters but in this case you

want to clear decimals so either you'll multiply by 10 by 100 by a thousand just depends how many times do you want to

move the decimal if you multiply by 10 the decimal will move one time multiplying by 100 the decimal will move

two times and by a th the decimal moves one two three times in our case we only have one decimal place everywhere so we

need to move our decimal place by one place and therefore 10 would be the easiest thing to

multiply so times 10 to each term 10 * this time 10 and that will move the decimal by one spot in this case giving

us 4X minus8 the decimal moves by one spot

giving us eight and 1.2 the decimal moves by one spot giving us 12 so after one step of fraction Busters in this

case times 10 the decimals are cleared all right on to regular solving add and subtract before multiplying and dividing

so Plus 8 we have 4X left over on that side and 12 and 8 is 20 last step divide by 4 to get X alone and now x equals

5 done so to practice more go to YouTube Type in either search phrase here play the video but pause it so you can try

the problems yourself and then continue the video after you did it yourself to see if you are correct all right number

seven and eight number seven solve the inequality so first solving

these is just like solving a regular equation with an equal sign nothing's different except for one thing at the

end if you end up with a u like -2X is less than say let's say eight if you divide a negative to an inequality

then that's the time to remember you must reverse the inequality and same thing if you

multiply a negative to both sides of inequality you also reverse the sign otherwise the sign stays the same side

or sign same direction and you just solve as if it was a regular equation all right let me clear this out and then

do this problem all right uh let's make a mistake at the very beginning same as

before if you're thinking minus one to both sides that's a good idea but this is not really one it's actually three

times all this stuff so it's really3 so subtracting one isn't really the truth this one is actually a neg3 because it's

neg3 times all the stuff so really our first step has to be to distribute so let's distribute the

parentheses so-3 * 2 is -6x -3 * 1 is -3 and still greater than 7 all right now that we distributed we

can do regular solving for x let's combine like terms 4x - 6X is -2X still Min -3 and still greater than

7 before dividing it's easier to add and subtract to both sides so in that case We'll add three to cancel out

the3 we're left with -2X is greater than 7 + 3 is 10 and finally divide by2 and since we're dividing by a negative we're

going to reverse the symbol so go X is less than5 if I were to graph this it didn't

say graph it but we're going to anyway here's a number line here's zero here's five here's neg

five since it says X is less than5 there's an open circle at neg5 because you can't equal neg5 it didn't

say this or equal to it just says less than only so open circle for less than and then which direction is smaller than

or less 5 that is to the left because -67 and so on so you would shade to the left hand

side if you want to practice more of these go ahead and type in these Search terms into YouTube pause whatever

problem the person's doing try it yourself and then resume the video to see if you are

correct okay number eight this is called the compound inequality because it's kind of two equations of one it's this

equation saying that -2 is less than 2x + 2 and at the same time it is also I'll switch to Blue it is this equation

saying that at the same time 2x + 2 must be less than 8 so it's really two equations in one and you can solve it

either all in one shot or as separate equations I'm going to solve it as two separate equations and then at the end

I'll go back and solve it as one equation so here's the separated version all right knowing that this is two

equations then let's solve them separately and at the end combine them on the left hand side we'll subtract two

to both sides that leaves us with 2x on the right side -2 - 2

is4 divide two because we're dividing by a positive the inequality stays the same direction we're not dividing by a

negative right this isn't negative -2 or dividing by it's positive two so the symbol stays the

same4 ID 2 is -2 so there's one side and at the same time not only does X have to be larger

than -2 but once we solve this we'll see the other side of X those cancel out 2x is less than

6/ two and then on this side x has to be less than three so the solution is this x has to be less than three and at the

same time also larger than --2 that's how you'll see it written it means both at the same time on a number line Looks

like this here's 0 1 2 3 1 -23 there's an open circle at three

because X is less than three there's an open circle at -2 and X has to be less than three but

also larger than -2 so you only shade the part between these two so X can be any number between but it can't go

beyond this this range so that's the way to split up the equation to two and here's the way to do this as one

shot delete stuff at one time all right here's a repeat when solving this kind of equation you can

treat it as like a three-part equation subtract two from all you know all three sides left middle right those cancel out

so 2x is less than 6 on this side -2 - 2 is -4 finally divide by two those cancel out and then you got X

is less than three on one side but also larger than -2 here and that gave us the same thing we had here and the graph

would be the same as well so to practice more you want to search up solving compound inequalities on YouTube Cube

and you'll have more practice problems all right number nine is xals ne4 a solution to this

inequality all we need to do is substitute -4 in place of X and find out is it a true statement or false because

if4 is a solution when you plug it in it should make a true statement so let's find out 18 - 2 * -4 is it true that

that is less than 26 I don't know let's just keep calculating all right -2 * -4 is POS 8 is that less than

26 sorry that's like I said positive 8 18 + 8 is 26 is it true or false that 26 is less than itself and that is false 26

is not less than itself so this is false and therefore the three dots is is the math symbol for the word therefore

therefore x = -4 is not a solution because it doesn't make the

equation true Solutions are values that make the equation true all right to practice more problems search up this in

YouTube and you'll get more problems to practice and go from there number 10 this is known as function notation so

this one a bunch of you really should just watch some videos on function notation first first and then come back

to this video all right so here's my brief introduction so in function notation it's a machine where you input

values and get outputs in this case let's say I wanted to find out so f is the name of the function it's function f

and X is the input so if I input let's say I don't know three into function f I would have 6 - 5 * 3 so this part

represents the input and this is the equation name that's the name of the function and this entire

thing gives you the output like stands for the output so in this case if I input

three then the answer or output is 6 minus let's do up here 6 - 15 that

is 9 so if I input -3 I get an output of9 and you would read this as F of3 is9 and it

means if you input three to the equation F you get ne9 as the answer and so this whole thing F of three that's the answer

of9 when you plug in three all right on to our actual problem so for our problem they were

asking us to plug in zero and so that's what we'll do erace my

stuff all right find F of zero so let's input zero into equation F so 6 - 5 * 0 that cancels out it's just zero and we

get F of 0 is six so when we input zero we get six as an answer and therefore F of Z is six um the answer is six when

you input zero for more problems definitely go practice function notation here and because some

problems are backwards in fact let's do one now honestly here here's here's a backwards problem

um okay if f ofx equals let's say 10 find X so this is a backwards problem they're telling you if you get 10 as the

answer to the output what was the input that matched it and so to solve this kind of problem you would do this

backwards and basically put 10 on this side right 10's the answer the question is what's the x

value that gives you 10 so you would solve this equation we have here before us solve this for X to find out what x

gives you that so in our case we' subtract six to both sides 10 - 6 is

45x let me rewrite this over here divide by5 and so the answer is x = -4 FS I

made it up so we got fractions but basically there's two directions either you input something for x or they give

you the answer and you have to find X and I think that if you search the videos you're going to find both those

kinds of questions so right here so yeah check out function notation on YouTube and find lots of

practice problems until you're 100% confident in those questions all right number 11 is the graph at the right a

function so here's what to know for functions U the rule is each

input which is X has to have um only one output or Y so in this graph does each x value

have only one y here let's look at like xal 10 for xal 10 right the graph it lives up here and the graph exists down

here so for an X of 10 y could either be this height or this height that's two different yv values for One X value so

already this is not a function not a function and it's because some X values have more than one y-value

a different way to do this is to use the vertical line test so vertical line test if you pass a

vertical line through the equation or through the graph if it touches at more than one spot vertically that means it's

not a function because it means that for a certain X Y has two possibilities or maybe more like this graph right here

that would have three possible y- values for One X value and that's definitely not a function

and for the graph we were given it doesn't pass the vertical line test because even though it passes right here

right that would just be one x value but it doesn't pass all these lines and therefore not a

function um if this have been sideways by the way or that way so if this graph had been this a parabola a parabola is

the name of the shape if this paraba had been opened vertically then the vertical line test would pass because at each

time that you draw a vertical line it only crosses once on the graph so this would be a

function all right let's do number 12 let me erase my drawings okay domain arrange so as a

review domain is the set of possible so possible X values that the graph covers and the

range is the the possible y values that the graph covers so in this

case what x values does the graph cover right it goes from a x of five right here and then it goes over to the right

forever really I'm assuming there's arrows on here they didn't draw them but I think it was meant to put that way so

this case X goes from five over to the right forever so the domain is X is greater than or equal to

5 because the graph covers X values from five over to the right forever okay for y

values what's the highest and lowest this graph ever goes right I can see that a y of zero is right here that's a

y of zero yals 1 is this height and the graph exists for that height too height of two is are here the graph

also exists for height of two so y it goes from zero up forever and it goes from zero going down forever

so this graph eventually covers all values up it covers zero and it covers all values going down so y the range is

all real numbers all real numbers and if you want you can put Y is an element of the set

of real numbers this is math notation a Double stemed R is the the symbol for real numbers which are decimals

fractions and just all the numbers that are possible and this little other symbol means element of so why is

everything all right for more practice go ahead and search up these Search terms in YouTube and you'll get more

practice problems almost there number 13 write an equation of a line in slope

intercept form so that's the first thing here's what slope intercept form is it's the classic one y = mx plus b because

you're given a slope and intercept that's why it's called slope intercept form so to do this problem we need to

know two things the slope and intercept so let's find the slope by taking our two points and finding the rise over

run using this equation Y2 minus y1 that'll find the rise between the heights and then X2 2 - X1 we'll find

the run so let's label these that's X1 y1 it's our first XY point and the other one will be our X2 and Y2 in other words

0 2x and0 2 is y all right so second y-coordinate minus the first one that would be

4 minus 2 divided by the X values will be 0-1 make sure you catch that it's 0

minus a negative 1 4 - 2 is 2 0 - NE 1 is positive one because negative negative is positive and then two over

one is the same as two so at the end we've got this we know y equals a SL of 2 * X plus some Y intercept I'm not

quite sure yet all right to find the Y intercept there are two ways here you could do one if you can see already

they're giving us the Y intercept 0 comma 4 is the Y intercept so instantly we know that this is

four because Zer comma anything is a y intercept so we're done let's just say for the sake of argument so this is the

final answer by the way but let's say that it wasn't 04 and it was I'll make something up let's say the answer or it

gave us like 2A 6 so what line would go through let's say had the same two but it had 2 comma 6 instead and not 04 in

that case here's what you would do here you would just have to put plus b pretending like we don't know what B

is and you could then plug in the two and six for X and Y to find out what B is so X would be two so 2 * 2 y would be

six and then you would solve this for B so 2 * 2 is 4 you would subtract four from both sides

get myself some more space here subtract four subtract four that's two so then you have b equals

2 and the equation in this case would be yal a SL of 2 x + 2 so if you don't know the B because they don't give it to you

you can still plug in one of the points for x and y and solve for the B that wasn't needed here all right for more

practice of course you can type in this search phrase into YouTube and you'll get a ton more practice

problems all right let's do number 14 so 14 find the equation of a line in slope intercept form that's still MX

plus b that is parallel to this line so parallel means same slope but different y intercept

different Y intercept so we know the slope of our new line definitely has to be two so so far we know this our new

line to be parallel has to be y = 2x to be same slope but then plus a different Y intercept and we want our line to go

through Z 0 but that's already the Y intercept because Z comma something right zero comma whatever is a y

intercept so in our case they're giving us the y intercept of Z so we have y = 2x plus a y intercept of 0 that doesn't

really matter so you have like yal 2x is the solution done but same thing let's say this point was not 0 and they're

like hey find a line parallel to this that includes the point I don't know 8 comma 6 okay in that case what you would

do is you would start here and you would plug in a comma 6 and find B so I'll do that in a second

here so if you're not given the Y intercept just outright as we were then you can always find B by plugging in our

case let's plug in eight for x and six for y so six is Y 2 * 8 forx plus some intercept we're going to find out what

it is 2 * 18 or 2 * 8 is 16 subtract 16 to both sides and now you

can see that 6 - 16 is -10 so our Y intercept B is -10 and therefore the equation that we parallel to this one

but goes through 8 comma 6 would be yals same slope of two but different Y intercept of -10 so that would be the

answer if they said it goes through 8 comma 6 and again to find more practice problems type this into YouTube find a

video and then pause the video try those problems yourself then resume the video and see how close you got to the answer

all right final question so there are two ways to do this you can use either technique of the

substitution method or elimination I will do both because sometimes one's easier than another but you need to know

both so let's say substitution method on this side and then elimination method

on this side okay let me write down the equations over

again so for substitution let's just start with the original problem 2x - 3 Y is 7 and then x + y is six same thing

over here I just need to rewrite the problems all right so for substitution method you need to get one variable

alone so you need to get either x equals or you need to get y equals by itself once you have that you substitute into

the other equation so for us it looks easiest to solve the second equation for x or y because there's no dividing it's

just regular X and Y so I'm going to choose to solve let's call this equation one call that equation two I'm going to

solve this for x by subtracting y to both sides and then we get

X = just 6 - y so this is our new equation and now for the substitution technique itself since x equals or is

the same as 6 - y we can substitute this in place of X in the other equation so that is the substitution method so

equation one after substituting becomes two times but instead of X it's going to be 6 - y

y but still - 3 y = 7 so basically I just rewrote this first equation over again except for in place of x i

substituted 6 - Y in place of the X and that is substitution all right now just solve regular distribute to the

parentheses 2 * 6 is 12 2 * Y is -2 y still - 3 y still equal 7 come my in terms -2 y - 3 Y is -5 Y and still 12us

that equals 7 I'll continue over here so I

have 12 - 5 y = 7 subtract 12 from both sides we got - 5 y = 7 - 5 or 7 - 12 is5 divide by5 to both sides and we end

up with Y = 1 okay once you find out one variable in this case Y is one then just plug that back into either equation to

find the other one so I'll use this equation because it looks the easiest to plug into so we got x + 1 for y =

6 and then subtract one to both sides x = 6 - 1 is 5 so now I found both X is 5 and Y is 1 so this is the point where

the two lines intersect each other and that is the solution to the system here's the same thing but using the

elimination method with the elimination method you basically want to add the equations vertically to force either X's

to cancel out or y's to cancel out but in this case nothing cancels because 2x plus X if you add them vertically that

gives you 3x if you add -3 y + y that gives you minus 2 Y and nothing cancels so this case you

would have to multiply one of these equations by something to force either X or Y to

cancel so if you want to cancel the X's you can multiply this whole equation by -2 to force a -2X here that would cancel

another choice you could also multiply by positive3 to force the Y to cancel if you want a three there but I'm going to

choose the X's so after multiplying the second equation by -2 we get this equation so top equation stays the same

minus 3 y still equals 7 the bottom equation -2 * X is -2X -2 * postive Y is -2 Y and -2 * 6 is

-2 so now if you add the equations vertically the X's cancel out -3 - 2 y is5

y 7 - 12 is5 and we get the same thing as we had before so yal 1 plug that back into to either equation and you're going

to get an X of five and you get the same solution as before so that's the elimination

method uh that's it so to practice more on YouTube just search up the key terms you can see below

for either the elimination method or substitution and you get tons more practice props just pause those videos

try them yourself and you've got infinite practice there all right that's it for me have a great day wish you the

best of luck on finals um see you next time