SAT Math Test Prep Online Crash Course Algebra & Geometry Study Guide Review, Functions,Youtube

today i'm going to focus on the math

section of the sat

so i'm going to go over six lessons the

first lesson is on algebra solving

equations evaluating functions including

composite functions and multi-variable

functions and then we'll move on to

lesson two converting sentences into

equations

including solving a series of word

problems

and lesson three ratios proportions

probability lesson four averages

fractions percentages

less than five graphs of linear

quadratic and absolute value functions

including slope arithmetic and geometric

sequences

and then lesson six a review of geometry

now in each of these lessons i'm going

to start out with a review of the

important topics equations and concepts

that you need to know and then we're

going to work on a series of multiple

choice problems

now for each of these problems i want

you to try to solve it

first before looking at the solution so

make sure you pause the video

do the problem first and then check your

answer by unpausing the video and

watching the solutions

this is the best way to get the full

benefit from this video if you want to

do well on a math section of the sat so

let's begin

let's start with the first lesson on

the algebra part on solving equations

evaluating functions

factoring and things like that let's go

over the basic concepts that you need to

know and then we'll work on a few

multiple choice problems

so let's start with exponents

when you multiply

common bases

you are allowed to add the exponents so

x cubed times x to the fourth power

is x to the seventh power three plus

four seven

whenever you raise one exponent to

another exponent

you can multiply it so five times four

is twenty

and whenever you divide

one base by another common base you can

subtract the exponents so nine minus two

is seven

now let's say if you have a radical

the cube root of x to the fifth power

how can you convert that into a

fractional exponent

this is the same as

x raised to the five thirds so likewise

let's say if you have the seventh root

of x to the nine you can rewrite this as

x raised to the nine over seven

now let's say if you wanted to evaluate

four raised to the third power what does

that equal

now four cubed means that you're

multiplying

three-fourths

four times four times four is 64.

likewise if you want to find the cube

root of 64 you need to find what number

times itself three times is 64. so what

times what times what is 64 which we

know it to be four

so let's say if you want to find the

fourth root of 16

what times what times what times what is

16 this is 2

because 2 to the fourth power which is

two times two times two times two

that's sixteen

now let's say if you wanted to simplify

a fractional exponent let's say if you

have eight raised to the five thirds how

can you find the value of this term or

of this

this number

what you could do is you can separate

the fraction

into two numbers

five thirds is the same as one-third

times five

so eight raised to the five-thirds is

the same as the cube root of eight

raised to the fifth power

eight to the one-third or the cube root

of eight is equal to two because two

times two times two three times is eight

and now we want to find out what two to

the fifth power is

so two times two times two times two

two times two is four this is four and

this is two

four and four is sixteen so sixteen

times two is thirty two so two to the

fifth power is 32.

let's try another example

so let's say if you want to

find the value of 16 raised to the 5 4.

the first thing you want to do is find

the fourth root of 16

and then raise to the fifth power

the fourth root of 16

is two and two to the fifth power we

know it to be 32

so let's say if you want to solve an

equation that looks like this x raised

to the two thirds is equal to 16. how

would you do it

in order to solve for x you need to

convert

the exponent from two thirds and change

it to one the only way you can do that

is to raise both sides to the three over

two

two thirds times three halves is one

and so we gotta find out what sixteen

raised to the half

raised to the third is

16 to the half is the same as the square

root of 16 and the square root of 16 is

4

and 4 raised to the third power

we know it to be 64.

so now let's say if you have a square

root the square root of five and you

wish to square it what is this equal to

the square root of five squared is five

the square and the radical cancels

because the index number is a two

it's always assumed to be a two if

there's nothing there

you can also see it this way the square

root of five squared is basically

radical five times radical five five

times five is 25

and the square root of 25 is 5.

so if you were to see the square root of

8 squared this is simply equal to 8.

now let's talk about the absolute value

the absolute value of a positive number

is a positive number the absolute value

of a negative number

is also a positive number

so with that in mind

how would you solve for x

in this equation the absolute value of x

plus one is equal to three

so what's the answer for x

whenever you have an absolute value

equation you need to write two equations

x plus one can equal to positive three

and it can also equal to negative three

because the absolute value of 3 and

negative 3 is the same positive 3.

so in the first equation if you subtract

by 1 from both sides the first answer is

2

and for the second one negative 3 minus

1 is negative 4.

so you get two answers for an absolute

value equation

so now let's move on to fractions

if you want to add five plus two-thirds

how can you do so

five is the same as five over one

whenever you want to add or subtract

fractions you need to get a common

denominator so the first fraction let's

multiply top and bottom by three

whatever you do to the top you have to

do to the bottom

so five times three is fifteen three

times one is three

so now that we have common denominators

we can add the numerators 15 plus 2 is

17

and so the answer is 17 over 3.

now

what if we wish to multiply uh two

fractions

if you want to multiply two fractions

you multiply across three times five is

fifteen

and uh two times six is twelve and now

we can reduce it

let's divide top and bottom by three

15 divided by three is five 12 divided

by three is four so the answer is five

fourths so when you're multiplying

fractions

you can multiply across

what about if we

wish to divide

two fractions

there's something called keep change

flip you can keep the first fraction

change division to multiplication and

flip the second fraction

so 8 times 3 is 24 and 4 times 5 is 20.

so let's divide top and bottom by 4 to

reduce the fraction

24 divided by 4 is 6 20 divided by 4

is 5 so the final answer is 6 over 5.

now let's say if

we have a fraction with large numbers

now we can multiply across however we're

going to get bigger numbers and then

we'll have to reduce the fraction

so if you're dealing with large numbers

simplify first before you multiply

so 8

we can rewrite eight as four times two

and twenty is four times five

fifteen is five times three

and twelve is four times three

so notice that we can cancel a five

and we can cancel a 4

and we can cancel a 3.

so right now what we have is 2 divided

by 4

which we can reduce that further if we

divide both numbers by 2.

so then the final answer is one-half

so we don't need to know what 8 times 15

is or 20 times 12.

so if you're dealing with fractions

simplify first before you multiply if

you're dealing with large numbers

you can do so for small numbers too but

for small numbers you don't really need

to but for large numbers you can solve

faster if you simplify it first

so now let's say if you have two

fractions

separated by an equal sign how can you

solve for the value of x

whenever you have two fractions

separated by an equal sign

you can cross multiply

so three times four is twelve and five

times x plus two is five x plus ten

so solving for x let's subtract both

sides by ten

so two is equal to five x and if we

divide both sides by five x is equal to

two over five

so that's when you can cross multiply

so now let's move on to factoring

let's say if we have a trinomial where

the leading coefficient is one

the leading coefficient is the number in

front of x squared

if you want to factor an expression like

this

find two numbers that multiply to 15

but that add to the middle term eight

so one in 15 multiplies to 15 but as is

six i mean sixteen

the other option is three and five three

times five is fifteen but three plus

five is eight

and so when we factor it's going to be x

plus three

times x plus five

and because factoring is a common

technique that is uh needed on the sat

let's do a few examples

so go ahead and factor this expression

so find two numbers that multiply to 28

but add to 11. so let's make a list 28

divided by 1 is 28 if we divided by 2 is

14

3 doesn't go into 28 but if we divided

by 4 is 7 notice that 4 plus 7 is 11.

so it's going to be x plus 4 times x

plus 7.

so go ahead and try this example x

squared

plus 3x

minus 21.

so let's make a list of the factors of

twenty one

so we have one in negative twenty one

two doesn't go into it

three and negative seven

actually this one is not

factorable so let's change it

let's make it uh x squared

minus 4x minus 21

actually plus 4x minus 21.

so notice that 3 plus negative 7

is negative four but if we change it to

negative three and positive seven it now

adds to positive four

so this will be x minus three

times x plus seven and that's how you

would factor it

try this one

x squared minus 9x plus 20.

so what two numbers multiply to 20 but

add to negative nine

so if they're adding to a negative

result

um we need two negative numbers if

they're going to multiply to a positive

number

so this could be negative 1 and negative

20 negative 2 and or negative 10

negative 4 and negative 5

but these two add to negative 9 so it's

going to be x minus 4 times x minus five

but now let's say if the leading

coefficient is not one

let's say if it's

a two

what can we do under these circumstances

the first thing we need to do is

multiply the leading coefficient by the

constant term

so 2 times negative 2

that does not look like a negative 2

is equal to negative 4.

so you need to find two numbers that

multiply

to negative four but add to the middle

term negative three

so this has to be negative four and

positive one negative four times one is

negative four but negative four plus one

is negative three so now what we need to

do is we need to replace the middle term

with negative 4x

plus 1x

notice that negative 4x plus 1x still

adds up to negative 3x so the value of

the expression is still the same

it's just

written in a different way

so now at this point what you want to do

is you want to factor by grouping

so you want to take out the gcf the

greatest common factor in the first two

terms

the gcf is 2x

now to find out what's left over on the

inside divide 2x squared divided by 2x

is x and negative 4x divided by 2x is

negative 2.

so now if we take out a 1

it's just going to be x minus 2 on the

inside

if these two factors are the same then

you know you're on the right track you

haven't made any mistakes thus far so

now we're going to take out x minus 2.

if we remove x minus 2 from this term

the only thing that's left over is 2x

and if we remove x minus one from this

term the only thing that's left over is

plus one and so that's how you factor it

so let's try another example where the

leading coefficient is not one

try this one

six x squared

plus seven x minus three

so if we multiply six and negative three

that is equal to negative eighteen so we

need to find two numbers that multiply

to negative eighteen but add to seven

so we have one and negative 18

2 and negative 9

3 and negative 6. notice that 2 plus

negative 9 is negative 7 but if we

change it

negative 2 and positive 9 adds up to

positive 7.

so let's replace

7x with 9x

minus 2x

the order doesn't matter we could make

it negative 2x and 9x

so now let's factor the gcf the gcf for

the first two terms is 3x

so 6x squared divided by 3x is 2x and 9x

divided by 3x is 3.

now if we factor out a negative 1 for

the last two terms negative 2x divided

by negative 1 is 2x negative 3 divided

by negative one is plus three

so because these two terms are identical

we know we are on the right track

and what goes in the next fraction is

what we see on the outside the three x

and a negative one

and so that's how you can factor

an expression or a trinomial where the

leading coefficient is not one

so now let's talk about

some other functions

let's say if you want to factor x

squared minus 25 how can you do it

right now this is in the form of a

difference of perfect squares

so a squared minus b squared can be

factored into a plus b

and a minus b

so what you really need to do is take

the square root of x squared which is x

and the square root of 25 which is 5

one side is going to be positive and the

other side is going to be negative

so let's say if you want to factor 4x

squared

minus 81.

so what's the square root of 4x squared

we know the square root of x squared is

x and the square root of 4 is 2 so

for 4x squared it's going to be 2x and

the square root of 81 is 9.

so one is going to be positive and the

other is going to be negative

now

let's say if you want to square

factor excuse me 9x squared minus 64 y

is to the fourth

so what's the square root of 9x squared

that's going to be 3x and the square

root of 64. we know it's 8 and the

square root of y to the fourth what you

basically do is take the exponent

divided by 2 so it's going to be y

squared so it's going to be 8y squared

and on one side it's going to be

positive and the other side is negative

so that's how you can factor using the

difference of perfect squares method

so now there are some other things that

you need to know

let's say if you see an equation in this

form a squared plus two a b

plus b squared

this can be factored as a perfect square

a plus b squared

likewise if you were to see a squared

minus two a b

plus b squared this is equal to a minus

b squared

so let's see if you get a question

and they give you something like this r

squared

minus 2 rs

plus s squared is equal to 49

what is the value of r minus s

notice that r squared minus 2 rs plus s

squared that's equal to

r plus s squared if you

were to factor it so therefore

actually not r plus s squared r minus s

squared

it has to be negative

so if you factor it will be r minus s

squared so if you want to find r minus s

you simply have to take the square root

of 49 which is 7.

so let's say

if you have this equation x squared

plus

well actually let's say if you have x

plus y

is equal to 5

and you want to find the value of x

squared

plus

2xy plus y squared

you want to know what that's equal to

well if you factor this expression using

the equation that we mentioned in the

last page

this is equal to x plus y squared

so basically you just have to square 5.

5 squared is 25

and so you might see a few questions

like this on the sat sometimes you have

to square it sometimes you have to

square root it you just got to know when

you have to do what in each case

so now let's talk about

solving equations that have fractions in

it so let's say if you have 5 plus 2

over x

and that's equal to 1 how can you solve

for x in this equation

what i would recommend doing is multiply

both sides by the common denominator so

there's only one denominator here so

let's multiply by x

so you got to multiply everything by x 5

times x is 5x

2 over x times x

the x variables cancel so you're left

over with two

and one times x is x

so now what we're going to do is we're

going to subtract both sides by x

and subtract both sides by 2.

so these they will disappear

5x minus x is four x and not equal to

negative two

so if we divide by four we're going to

get negative two over four which is

negative one half and that's how you

could solve

for x it's gonna be a lot easier if you

multiply both sides by the

denominator so let's try another example

like that let's say if we have 3 over 2

plus 4 over x

and let's say that's equal to 3.

so here we have two different

denominators 2 and x so let's multiply

both sides by the common denominator

which is two x

so what's three over two

times two x which is the same as two x

over one

notice that the twos cancel and what we

have left over is three times x which is

3x

and if we multiply 4 over x times 2x

the x's will cancel and what's left over

is 4 times 2 which is 8.

so 2x times 3 over 2

we know it to be 3x the 2's cancel

and uh 2x times 4 over x the x's cancel

and we have 4 times 2 left over which is

8

and then finally 2x times 3 is 6x

so you have to multiply by um you have

to multiply 2x by every term

if you miss one term

your answer will be incorrect it's going

to be wrong

so make sure you multiply everything in

the equation by 2x so now we can solve

for x if we subtract both sides by 3x

so therefore 8 is equal to 3x and if we

divide by 3

x is 8 divided by 3.

so now let's move on into functions

let's say that f of x is equal to three

x plus five

if we wish to find the value of f of two

how can we evaluate it all we need to do

is replace two for x

so three times two is six plus five

that is equal to 11.

so we could say that f of 2 is equal to

11.

f of x is equal to y

if x is the only variable inside of the

function so as you can see 2 is the x

value the value on the outside is the y

value

so using the same function

f of x is equal to 3x plus 5.

what is the value of x

if f of x is equal to 29

how can you find a value of x

so we need to realize is that

y is equal to 29 so therefore the 3x

plus 5 the outside part is equal to 29.

and so you set the whole thing equal to

29 and then you can solve for x

subtracting both sides by 5 29 minus 5

is 24 and if we divide by 3

24 divided by 3 is 8 so x is eight

so keep in mind if the value is on the

outside you could set the whole function

equal to 29 if it's on the inside you

need to plug in for x

now sometimes you might have a function

that has two variables let's say if you

have f

of x comma y is equal to x squared plus

two y

what is the value of

f comma three

f of three comma five so we could see

that x is three and y is five

so therefore

it's going to be three squared plus two

times five three squared is nine two

times five is ten so the whole thing has

a value of

nineteen but now let's say if

f of 4

comma y is equal to 28 what is the value

of y

so notice that in this problem x is

equal to 4

and the entire function is equal to 28

so we can say that 28 equals x squared

plus 2y

and we know that x is 4 so we can plug

in 4 for x

and let's make some space

so therefore

4 squared is 4 times 4 which is 16 and

if we subtract both sides by 16 28 minus

16 is 12

and if we divide both sides by 2 12

divided by 2 is 6

so y is 6.

so that's how you can solve for a

variable if you have a function with two

variables

so now let's talk about composite

functions

let's say that f of x is 2x plus 1

and g of x

is equal to x squared

what is the value of f of g of 2

a composite function

is basically two functions where one

function is inside the other function in

this case g is inside of f

so what you should do is start with the

the part that's on the inside and then

work your way towards the outside

so let's evaluate g of 2.

so g of x is x squared so g of 2 is 2

squared which is 4.

so because g of 2 is equal to 4 i can

replace g of 2 with 4. so now i'm

looking for f of 4. so i'm going to plug

this into the function for f so 2 times

4 plus 1

2 times 4 is 8 a plus 1 is 9 and that's

the final answer for this

composite function

so

we've covered a few basic topics that

you'll need for the first lesson

in this sat course

so

we've covered factor in solving

equations

adding subtracting multiplying fractions

composite functions and basically

most of the stuff that you'll need for

the algebra part of the test

so at this point let's begin with a few

multiple choice problems

number one

if f of x is equal to three x squared

minus five x plus x cubed

then f of four is equal to

so for all of these problems that we

encounter in this video

pause the video and try the problem

yourself

and then see if you can get the answer

if you do it that way you're going to

get the most out of this video

so always try each question before you

look at the solution but let's begin

if you want to find f of 4

all you need to do is substitute

x with four so everywhere we see an x

value

we're going to replace x with four

so now we just have to do some math

four squared that's four times four

which is sixteen

five times four is twenty

and four to the third power that's four

times four times four

four times four is sixteen and sixteen

times four is sixty-four

3 times 16 is 48

and 48 minus 20 that's 28

and 28 plus 64

is equal to 92.

so therefore d

is the correct answer for this problem

number two

if f of x is equal to x squared plus

seven x plus five

and f of x is equal to 35

then what is the value of x

so

what's the first thing that you would do

to solve this problem

now keep in mind

f of x

is equal to y

so the number that's on the inside

of f

is equal to the value of x and a number

that's on the outside is equal to y

so

when we see the equation f of x is equal

to 35

the number on the outside

is equal to y and we're looking for x

so we have the equation y

is equal to x squared

plus seven x

plus five

and f of x and y are the same thing

they equal each other and so now we can

replace 35

with y and now we gotta solve for x

whenever you see

an x squared

and an x variable with half the exponent

like x to the first power it's a

quadratic equation

and you may have to solve it either by

using the quadratic formula by factoring

or even by completing the square

so at this point let's subtract both

sides by 35.

so now we have zero

is equal to x squared

plus seven x

minus thirty

so now we need to factor this expression

what are two numbers that multiply to

negative 30 but that add to seven

the two numbers are

positive 10

and negative three

10 plus negative 3 is positive 7 but 10

times negative 3 is negative 30.

so in this factored form it's going to

be x plus 10

times x minus 3.

so now

what we need to do is set each one

equal to zero

so if we set the first factor equal to

zero

x plus ten equals zero

we could subtract both sides by 10 so x

is equal to

negative 10. so that's one possible

answer

the other answer x minus 3 is equal to 0

if we add 3 to both sides x is equal to

positive 3.

so

out of all the answers that are listed

3 is the right answer answer choice c

so that's it for this problem

number three if three x plus eight

is equal to twenty-four

what is the value of seven x plus three

so how can we figure this problem how

can we find the value of seven x plus

three the best way to do this

is to solve for x in the first equation

and then plug in the value of x in the

second expression

so let's start with the first equation

3x plus 8

is equal to twenty four

so let's subtract eight from both sides

so therefore three x

is now equal to

twenty four minus eight which is sixteen

so to solve for x we need to divide both

sides by 3.

therefore

x is equal to 16

over 3.

so now we want to find the value

of the expression

7x plus 3.

so what we need to do is insert the

value of x

into this expression

so

right now we have 7 times 16

divided by three

plus three

to add these two terms

we need to get common denominators

so we're going to multiply the second

term

by three over three

whatever you do to the top you have to

do to the bottom

so that the value of the fraction

remains equal so 3

is equivalent to 9 divided by 3.

now we need to know what 7 times 16 is

7 times sixteen

is one twelfth

so we have one twelfth over three

plus nine over three

and if we add those two

one twelfth plus nine is one hundred

twenty one

divided by three

so we can see that answer choice a

is the right answer for this problem

number four

if the square root of seven is equal to

x minus three

then x minus three squared is equal to

so how can we solve this particular

problem

now we could try the approach that we

used in the last problem and that is

solve for x in the first equation and

then plug it in into the expression on

the right into x minus 3 squared we

could do that

and that will work it will give us the

right answer

but we don't need to

if x minus 3

is equal to the square root of 7

then we could square both sides then x

minus 3 squared

must be equal to

the square

of square root 7.

now square root seven squared is simply

equal to seven

here's why

the square root seven squared

is the same as the square root seven

times the square root seven

this two on top means that we have two

square root sevens that are multiplied

to each other

the square root of seven times the

square root of seven is the square root

of 49 because seven times seven is 49

and the square root of 49

is 7

7 times 7 is 49 so

therefore x minus 3 squared is equal to

7.

so b is the right answer

now let's just see what would happen

if we solve it

uh using

uh the approach that we used in the last

problem

so starting with this expression

we could solve for x by adding 3 to both

sides

so 3 plus root 7

is equal to x

so now we can try to find the value of x

minus 3 squared

and so

since x is equal to 3 plus the square

root of 7 we could take that value

and insert it for x

so this is going to be 3 plus root 7

minus 3 squared

so the 3's will cancel

and then what you have left over is root

seven

squared which we know to be seven

so

both methods or techniques will work

so whichever technique

you feel comfortable with that's the one

that you should use

so b is the right answer for this

problem and let's move on to the next

one

number five

if 4x is equal to 12

what is the value of 3x minus 7 squared

so let's solve for x

so if 4x is equal to 12

we could find the value of x by dividing

both sides by 4.

so therefore x is equal to 3.

so now in order to find the value of 3x

minus 7 squared

we need to take the value of x and

insert it into this expression

so it's going to be 3

times 3

minus 7 squared

3 times three is nine

and nine minus seven is equal to two

and two squared is basically two times

two which is equal to four and so

therefore a

is the right answer for this problem

number six

if x plus four squared is equal to eight

x minus ten squared

then the value of x is

so how can we figure this problem

well let's take the square root of both

sides but first let's rewrite the

problem

so x plus 4 squared is equal to 8x minus

10 squared

so we need to take the square root of

both sides

when you square root a square you need

to keep in mind that the index number is

two

and so

the twos will cancel

and therefore the square root will get

rid of the square

and so we don't need the parentheses

anymore so what we have is just x plus 4

is equal to 8x minus 10.

now

what you need to keep in mind is that

whenever you take the square root of a

number

you can get a positive answer and you

can get a negative answer

so for the negative answer all we need

to do

is change one side of the equation

or multiply one side of the equation by

negative one

and then it's going to work

so let's start with the

equation on the left let's subtract

x from both sides

so

we're going to have 4 is equal to 7x

minus 10.

so next let's add 10 to both sides 10

plus 4

is 14.

so 14 equals 7x and then we're going to

divide both sides by 7

so 14 divided by 7 is 2.

so 2 is one possible answer but notice

that

it's not one of the choices so therefore

we can't really use two as an answer

so now let's work with the other

equation on the right so first let's

distribute the negative

to the right side so negative 1 times 8x

is negative 8x and negative 1 times

negative 10

if we distribute

that's going to equal

to positive 10.

so now

let's add

8x to both sides

and simultaneously let's subtract both

sides by four

so this will cancel and that will

disappear as well x plus eight x is nine

x

and ten minus four is equal to six

so next we need to divide

both sides by nine

so therefore we could see that x

is equal to 6 over 9

and if you divide

both numbers by 3

since they're both divisible by 3

you can get a reduced fraction 6 divided

by 3 is 2 9 divided by 3 is 3 so

x is therefore equal to 2 over 3 or 2

thirds so b is the right answer for this

problem

but let's check it let's prove that

this value is indeed the right answer

so let's plug in two-thirds for x

so two-thirds

plus four

squared should equal

to um

eight times two-thirds

minus ten

squared

so let's get common denominators four

over one is the same as

twelve over three if you multiply top

and bottom by three

you'll get 12 over three and 12 divided

by three is four so the value remains

the same

now eight times two thirds eight times

two is sixteen so we have 16 over three

and

10 over one what we could do to get

common denominators is to multiply ten

by three over three

so ten is the same as

uh thirty over three thirty divided by

three is ten

so now we can add the two fractions so

two thirds plus twelve thirds

is equal to uh fourteen over three

squared

and sixteen

minus thirty is negative 14 over 3

squared

so

on the left side

we have 14 over 3 times 14 over 3.

that's what 14 over 3 squared means

now on the right side we have negative

14 over 3 so that means that we have two

negative numbers negative 14 over 3

times negative 14 over 3.

in both cases

fourteen over three times fourteen over

three will be 196 divided by nine

and on the right side two negatives um

make a positive

negative fourteen over three times

negative fourteen over three is the same

answer in 196 over 9. so therefore

because the left side is equal to the

right side

this equation is true

so we can see why b

is the correct answer

so whenever you take the square root

just keep in mind you may have a

positive answer and you may have a

negative answer so you need to check

both to see which one is the right

answer

or which one is the answer that's listed

in this problem

number seven

if eight times the fourth root of x

cubed

minus 15 is equal to 49

then the square root of x minus four is

equal to

so you might see a lot of problems like

this on the sat where you have to solve

for x

in the first equation and then plug in x

to the expression on the right side

now as you can see the difficulty of

these problems are increasing

the main idea is the same

but

the steps that you need to take to solve

for x

might be different might be longer

sometimes it's easier

it can vary

but just make sure you know your algebra

you get you just you gotta know your

stuff

so let's start with number seven

let's rewrite the problem first

so the first thing we need to do

is we need to add 15 to both sides

so 49 plus 15

is equal to 64.

next we need to divide both sides by 8.

64 divided by eight is equal to eight

now

how can we rewrite the radical the

fourth root of x to the third

how can we rewrite it as

a fractional exponent

the fourth root of x cubed

is equal to x raised to the

three-fourths

so let me give you another example let's

say if

you have the seventh root of x to the

third

this is equivalent to

x raised to the three over seven

so now how can we solve for x

for this equation that we have at this

point

so we need to change the exponent from

three-fourths to one

because x is the same as x to the first

power or x raised to the one

so in order to change it

to one we need to raise both sides

to the reciprocal of three-fourths which

is four over three

and whatever you do to the left side you

have to do to the right side

so when you raise one exponent to

another you have to multiply for example

x cubed raised to the fifth power

is equal to x to the fifteen you

multiply three and five

so three fourths times four over three

the fours cancel

and the threes cancel

so three-fourths times four-thirds is

simply one

so we have x raised to the first power

is equal to eight

raised to the four-thirds

so now how can we find a value of

eight to the four thirds

what you wanna do is you wanna separate

the three and the four

eight to the four thirds is the same as

eight raised to the one-third

which is raised to the fourth

because one-third times four is

four-thirds

so the value of this expression is still

the same we just rewrote it in a

different way

so if you want to find out the value of

eight to the four thirds the first thing

you should do is take the cube root of

eight

the number on the bottom is the index

number that's the root

and the number on the top is like the

the exponent you're gonna

raise it to the fourth power but first

let's find the cube root of eight

the cube root of eight

is a number where

before i give you the answer here's what

you need to ask yourself if you want to

find the cube root of eight

find out what number

times itself three times is equal to

eight so what times what times what is

eight

the answer is two two times two times

two three times is equal to eight so

the cube root of eight is two

so now we gotta find out what two raised

to the four is

two to the four is basically two times

two times two times two

two times two is four

and the other two's on the right side is

also four so four times four is sixteen

so two raised to the fourth power is

equal to sixteen and therefore

um

that's not the final answer yet

so we need to avoid

the temptation of selecting an answer

when we're not finished yet

because i was about to do that

so what we now have is the value of x

x is equal to 16.

but our goal is not simply just to find

the value of x

we want to find a value of this

expression the square root of x minus

four

so let's take the value of x and insert

it into this expression

so the square root of sixteen minus four

the square root of sixteen

is four and four minus four is equal to

zero

so c

is the correct answer for this problem

number eight

if eight minus four over x is equal to x

plus four

which of the following is a possible

value of x

so

we just got to solve for x in this

problem

so let's begin

what's the first thing that

you think that we should do

how would you solve for x in this

expression

now the first thing that i would

personally do is i would try to

eliminate any fractions

before i try to solve for the equation

so notice that

the denominator of this fraction is x

so i'm going to multiply both sides by x

so

x

times 8

is equal to 8x

and 4 over x

times x

is equal to 4 because the x's

the x values they cancel so we're just

going to get negative 4.

and then x times x

is x squared

and x times 4

is 4x

so whatever you do to the left side you

have to do the same thing to the right

side

so we can't just multiply one side by x

and not do the same for the other side

so this is what we now have

notice that we have a quadratic

expression we have an x squared and an x

so whenever you see that what you want

to do at this point is you want to move

everything to one side

and try to factor the expression use the

quadratic equation or complete the

square to solve for x at this point

so let's subtract both sides by

8x

and let's add 4 to both sides

so this is zero right here

if there's nothing there it's it's a

zero

so these cancel so on the left side

there's nothing left over so it's a zero

so zero is equal to x squared and then

four minus eight is negative four

zero plus four is four

so now what we need to do is we need to

factor this expression

so what number what two numbers multiply

to positive four but add to negative

four

this is negative two and negative two

negative two times negative two is equal

to positive four but negative two plus

negative two is equal to negative four

so therefore in its factored form x we

have x minus 2 times x minus 2.

so if we set x minus 2 equal to 0 and if

we add 2 to both sides we could see that

x is equal to positive 2. so therefore e

is a possible value of x

now if you're having difficulty solving

for x

what you could do is you can

plug in each of these answers

and see which one

is true for the equation

so let me illustrate that technique

so let's say if you think

one is a possible answer

you can plug in numbers if you're having

difficulty solving for the equation

so eight minus four over one we're going

to replace x for one

is equal to one plus four eight minus

four is four and one plus four is five

so this is not true the left side does

not equal the right side so therefore d

cannot be a right um answer

so

now

let's try another value let's try e we

know the answer is e

so

let's replace x

with two

so four divided by two is two and two

plus four is six

eight minus two is also six so six

equals six

the equation is true so therefore we

know e is the right answer to this

problem

so you can always fall back to that

technique that is uh

basically looking at the answers and

plugging it into the equation to see if

it works

and

sometimes

that might be the best way to solve the

problem

it all depends on which technique is

faster

whichever technique can help you get to

the right answer quicker and that's the

technique you want to do because the sat

is a time test you have to be able to

solve the problem

very quickly

and accurately at the same time

number nine

if 4x minus 5y is equal to 6

what is the value of 16x squared minus

40xy

plus 25y squared

so how can we do this problem

now don't worry it might look difficult

but it's not

you need to be familiar with this

equation

a plus b squared

is equal to

a squared

plus two a b

plus b squared

so here's the proof

a plus b squared is the same as a plus b

times a plus b

so if you were to foil this expression

a times a

is a squared

and a times b

is a b

and then b times a is also a b

and then b times b

is b squared

so we can add the two terms in the

middle and that will give us uh

a b plus a b is two a b squared i mean

just two a b

so

therefore you need to realize that

a

is 4x in this problem

and b

is 5y

by the way

if there's a minus sign it's a minus b

squared

a minus b squared is a squared

minus 2 a b

plus b squared

but it's very similar so we can see that

a is equivalent to 4x

and b

is equivalent to 5y

so therefore

if a is 4x that means a squared is 4x

times 4x which is

16x squared

and if b is 5y that means b squared is

5y times 5y

which is

25y squared

so then the middle term is 2 times a b

so 2 times

4x for a

and 5y for b

so

4 times 5 is 20 times 2 is 40.

so this is negative 40xy

so how is this going to help us to get

the answer

so let's think about what this means

so

what this means is that 16 x squared

minus 40 x y

plus 25 y squared

is equal to

4x minus 5y squared

that's what we know

and our goal

is to find the value

of 16x squared minus 40xy plus 25y

squared we want to find out what this

what the left side is equal to

we don't know right now

but we need to use the right side to

figure that out

now we know that 4x

minus 5y is equal to 6.

so if that's the case we can replace 4x

minus 5y with 6.

so therefore

the left side is equal to 6 squared

which is equal to 36 and that's the

answer

so it's e

you just have to realize that

by squaring 4x minus 5y it equals to the

value of 16x squared minus 40xy plus 25y

squared so therefore all you have to do

is square 6 and you'll get the answer

so this problem is not hard

if you understand it once you understand

it getting the answer is easy all you

got to do is square 6 and that's it

you're done

but it's it's the understanding that you

need once you understand what to do

then math becomes easy

number 10

if r squared plus 2 rs plus s squared is

equal to 169 what is the possible value

of r plus s

so notice that

r squared plus 2rs plus s squared is in

the form

a squared plus

plus b squared

and we know that is equal to a plus b

squared

so make sure you understand how to

factor using this formula because it's

going to be very helpful when you're

taking your next sat exam

so this equation is true therefore

we know that r squared plus 2 rs plus s

squared is equal to

r plus s squared

and since r squared plus two rs plus s

squared is equal to 169

therefore r plus s squared is also equal

to 169 and our goal is to solve for r

plus s

so what we could do at this point is

take the square root of both sides

so on the left side we now have r plus s

which is what we're looking for

and the square root of 16 of 169 excuse

me

is equal to plus or minus 13.

so therefore

negative 13

is a possible value of r plus s

and that's the answer for this problem

so a is the right answer

number 11

if the product of x squared minus three

x minus ten and three x squared plus two

x minus one is zero

then x could equal

any of the following numbers except

so we're looking for

the values

that x cannot equal

so first let's convert the sentence into

an equation

so the product

of x squared minus 3x minus ten

product means multiplication

and we're gonna multiply this by

three x squared plus two x minus one

the product of these two terms

is equal to zero

or these two expressions

so we're not going to foil this

expression

that would be a terrible terrible thing

to do

what we should do is we need to factor

each expression

so let's start with the one on the left

so what two numbers multiply to negative

10

but add to negative three

so let's make a list

we have 1 and negative 10 and 2 and

negative 5.

2 and negative 5 works 2 times negative

5 is negative 10 but 2 plus negative 5

is equal to negative 3.

so

this is going to be x plus 2 times x

minus 5.

and if there's a one in front of x

squared once you get the two factors you

can simply write it um

in this uh in parenthesis if you get

these two numbers

now

for the expression on the right

the leading coefficient does not equal

one so we're gonna have to factor by

grouping

so there's gonna be a little bit more

work that's involved

for that part

so i'm going to factor it on the left

side first

the first thing you need to do is you

need to multiply the leading coefficient

3

and the constant term negative 1.

3 times negative 1 is negative three

and you need to find two numbers that

multiply to negative three but that add

to the middle term too

so what two numbers multiply to negative

three and add to positive two

try it

so this is none other than positive

three

and negative one

three plus negative one is two three

times negative one is negative three so

now what we're going to do is

we're going to replace the middle term

the 2x with positive 3x

and negative 1x

so that's all we did so far we replaced

2x with 3x minus 1x because 3x minus 1x

is still equal to 2x

it's simply expressed differently but

the value is still the same

so now we're going to factor by grouping

in the first two terms factor out the

gcf the greatest common factor

the greatest common factor

is 3x

you can take out a 3x from 3x squared

and 3x that's the most or the greatest

that you can factor out

when you factor out 3x from 3x squared

what's left over

to find out what's left over divide 3x

squared divided by 3x is x

and 3x divided by 3x is 1.

so now we're going to factor out

negative 1.

negative 1x divided by negative 1 is

positive x

negative one divided by negative one is

positive one

once you see that these two factors are

identical to each other you know you're

on the right track

so now we're gonna factor out x plus one

if we take out x plus one from this term

we have three x that's left over

and if we remove x plus one from this

term we have a negative one that's left

over

so therefore

the expression on the right can be

factored to x plus one times three x

minus one and all of that is equivalent

to zero

so now we can solve for x

so we can set each factor

equal to zero

if we set x plus two equal to zero x

will equal negative two all you need to

do is reverse the sign if x minus five

is equal to zero then x is equal to

positive five

and for this one it's negative one

now if three x minus one is equal to

zero

to solve for x we need to add one to

both sides

and then we need to divide by three

so x is therefore equal to one third

so those are the four possible values

for x

now we're looking for the exception

so we could eliminate answer choice a

we could eliminate uh b

we could eliminate c

and we could eliminate d

because we have those four answers

negative two one third negative one and

five

so the exception is e

x does not equal three um in this

equation

so therefore e is the right answer for

this problem

and the factorable expression

x squared plus kx plus 24

k is a positive integer

which of the following is not a possible

value of k

so

we need to find what value of k

will not allow this expression to be

factorable

so this is like a product sum type

problem

we need to find two numbers that

multiply to 24 but add to k

but how can we do that if we don't know

what k is

so first

let's make a list of all the

possibilities

all of all of the two numbers that

multiply to 24.

so

this would be 1 and 24

2 and 12. 3 and 8

4 and 6.

each of these pairs of numbers multiply

to 24. 2 times 12 is 24 3 times 8 is 24

4 times 6 is 24. so now what we're going

to do is we're going to add each of

these numbers

because k

is the sum 24 is the product the k is

the sum

1 plus 24 is 25

12 plus 2 is 14

3 plus 8 is 11

and four plus six is ten

so therefore we could eliminate

d because k could be equal to ten

six times four is twenty four but six

plus four is ten

and so if k was ten we could factor that

expression

if k was 11 we could factor it as well

and if k is 14 we can factor as well

however if k is 7

we can't factor it

if we had x squared plus 7x plus 24

this expression is not factorable

what two numbers multiply to 24 but add

to seven

we've already made a list of all the

numbers that multiply 24 and that's it

if k was 10

we could factor this expression

x squared plus 10x plus 24 would be

equal to

x plus 4 times x plus 6.

and so that's why 10 is not the answer

and because

there's no two numbers that multiply to

24 but add to 7

therefore 7

is not a possible value of k and so

answer choice a is the right answer for

this problem

13

how can we find the value of x

in this expression

so what's the first thing that you would

do

the first thing that we should do is we

need to factor

each expression

so let's start with the expression

on the upper left side

how can we factor x squared minus 2x

minus 24

the first thing we need to do

is we need to find two numbers that

multiply to negative 24

but that adds to negative two

and

so this is going to be six and four

but

which number is going to be negative is

it the six or the four

it has to be negative six and positive

four

negative six times positive four is

negative 24

and negative six plus four

is negative two

so

we can factor it or write it as x minus

six

times x plus four

now we need to factor this expression as

well

what two numbers multiply to 12

but adds a positive seven

this has to be four and three four times

three is twelve four plus three is seven

so

it's going to be x plus three

times x plus four

so now we need to factor x squared plus

x minus six

so what two numbers multiply to six but

adds a positive one

this is three and negative two

so this is going to be x plus three

times x minus two

so the first thing we need to do is we

need to simplify the expression

notice that we can cancel x plus four

and if we multiply the right side

and the left side

by x plus

three these terms will cancel

and the same is true for those terms so

what we now have left over on the left

side is simply x minus 6

and on the right side 12 divided by x

minus 2.

so now what we need to do at this point

is put this

over 1 and cross multiply

so 1 times 12 is 12

and we can foil

x minus 6 and x minus 2 if we multiply

those two

so foil in uh x minus six and x minus

two is going to be x squared

minus two x

minus six x and then six times two is

twelve so plus twelve

so at this point

we can combine like terms

negative two and negative six is

negative eight

and now let's subtract both sides by 12.

so therefore zero is equal to x squared

minus eight x

so we're going to factor out an x

so zero is equal to x

times

x minus eight

and so therefore

x can equal to zero and x can equal to

eight

any time you see an x on the outside

like this

x can equal to zero

so therefore

we're looking for a possible value of x

so e is the right answer

x could equal 8.

14

4b is equal to 64.

then the square root of b times the cube

root of 4b is equal to

so if 4b is equal to 64

then b is equal to 16 if we divide both

sides by 4. 64 divided by 4 16.

so now we can find out what the value of

this expression is equal to

so let's plug in 16 for b

the square root of 16 is 4

and 4 times 16 is 64.

and the cube root of 64

is

a number times a number times a number

that equals 64. and that's four four

times four times four three times the

64.

and four times four is 16.

so c is the correct answer for this

problem

fifteen

if x plus y is equal to eight

and x minus y is equal to four what is

the value of x squared minus y squared

so what we need to do is solve for x and

y and then we could plug it in to the

expression x squared minus y squared to

get the answer

so let's line up these two equations

and notice that we can solve it by using

the process of elimination

so if we add the two equations x plus x

is two x

y plus negative y is zero so they cancel

and eight plus four is twelve

so if we divide both sides by 2 12

divided by 2 is 6.

and so now what we can do is we can plug

in 6

into the first equation

so 6 plus y is equal to 8.

subtracting both sides by 6

y is equal to 2. so now we can plug in x

and y into this equation

so x squared minus y squared

that is equal to uh

six squared minus two squared

six squared is 36 2 squared is 4

and 36 minus 30 minus 4 is equal to 32

and therefore b

is the right answer for this problem

16

if 2x plus 3y is equal to 13 and 4x

minus 5y is equal to negative 7

then y minus x is equal to

so this problem is similar to the last

problem

so let's use elimination to solve it but

let's line up the two equations

so two x plus three y is equal to

thirteen

and uh

four x minus five y

is equal to negative seven

now

let's multiply the first equation by

negative two

so that we can get negative four x and

then we can add the two equations so

negative four x

and then three y times negative 2 is

negative 6y

13 times negative 2

is a negative 26. so let's add the first

these two equations

if we do that 4x and negative 4x will

cancel

negative 5 and negative 6 if you add

them it's negative 11 and negative 7

plus negative 26 is negative 33.

if we divide both sides by negative 11

y is equal to positive 3.

so now we can solve for x

using the first equation

so two x

plus three y or three times three since

y is three is equal to thirteen

so three times three is nine

and thirteen minus nine if we subtract

nine

on both sides

13 minus 9 is 4 and then 4 divided by 2

is 2.

so we have 2 for x 3 for y

so the expression y minus x is therefore

equal to three minus two which is equal

to one so one is the answer for this

problem

seventeen

if x times y is less than zero which of

the following must be true

so

in this problem let's try to disprove

every statement

the one that we cannot disprove

is the one that must be true

so

let's choose two values for x and y

such that the product is equal to a

negative number because the product

x y has to be less than zero which means

it has to be negative

so let's try positive 5 for x and

negative four for y

so looking at equation one

well first this must x y must be less

than zero so

five times negative four

is less than zero because negative

twenty is less than zero so five and

negative 4 works

is true for

this equation so now we can test

each of the choices to see which one is

true

so let's focus on x plus y

so if x is negative four and y i mean if

x is negative five and y is four

is it equal to zero

negative one does not equal zero

so therefore we have disproved number

one

so that means the answer can't be a and

it can't be c

so now

let's look at uh statement number two

three x minus three y

is less than zero

so three times five

minus three times negative four

let's see if it's less than zero three

times five is fifteen

and 3 times 4 is 12.

15 plus 12 is um

that's 27

27 is not less than zero

27 is greater than zero so that

statement is false so 2 is eliminated

because that means it can't be b and it

can't be d

so the answer has to be e

but let's see

let's try it just to make sure

so x squared plus y squared is greater

than zero

well

x squared

will always be a positive number and y

squared is always positive unless you

plug in zero if you plug in zero for x

then it's just zero if you plug in a

negative number for x like negative 2

when you square it it's going to be

positive

so we can see why number 3 is going to

be a true statement

a positive number plus a positive number

is going to be greater than 0.

and

we can't use 0 for x and y because

let's say if we chose zero zero

zero times zero

is not less than zero it equals zero

so

we can't use zero for x or for y

so which means x and y has to be a

number other than zero which means

number three will always be true

so if we plug in five

and negative four

we're going to get 25 plus 16

which is greater than zero

and let's say if we choose a different

test point

let's say um

x is a a negative number and y is a

positive number

negative three squared plus

five squared

will also give us a positive result this

is nine plus 25 which is also greater

than zero so we cannot disprove number

three which means three must be true so

therefore e

is the right answer

eighteen

if x is greater than zero which of the

following is equivalent to the square

root of x

to the fifth power

so let's go over

some rules associated with exponents

x squared times x to the third

is equal to x raised to the fifth power

when you multiply common bases you need

to add the exponents and x squared

raised to the third is equal to x to the

sixth power when you raise one exponent

to another exponent you need to multiply

the two exponents

so now we can answer the question

so the square root of x to the fifth

is equal to how can you write that as a

fractional exponent

keep in mind the index number if it's

not written it's always assumed to be a

two so this is

x to the five over two

so we need to find out which expression

is equal

to this one looking at number three

x raised to the half raised to the fifth

power

when you raise one exponent to another

exponent you gotta multiply so one half

times five is the same as one half times

five over one and that's five over two

so

number three is equivalent to uh this

expression

so

three is true

now let's look at number two

x to the fourth power times x to the

negative three over two

when we multiply common bases we need to

add the exponents

so

therefore we need to add

four

and negative three over two

so let's get common denominators let's

multiply this fraction on the left by

two over two so this is equal to eight

over two plus negative three over two

eight minus three is five so

this expression is equal to x raised to

the five over two so number two is a

true statement

so we can eliminate answer choice a

uh c

and d because they don't have uh number

three which we know to be true

so at this point we know two and three

is true so we can

we can clearly see that b is the answer

without even looking at number one

now number one is not true

let's say if you have x squared plus x

cubed this will not equal x to the fifth

power

you can't add unlike terms

so x squared plus x to the one half does

not equal x to the to the five over two

but let's prove it so

let's plug in a test point let's plug in

2 for x

actually now 2 let's plug in 4.

what is the value of 4 raised to the 5

over 2.

this is the same as 4 raised to the half

raised to the fifth power

so

anytime you raise something to the one

half it's like

finding the square root of that number

so the square root of 4 is 2

and two raised to the fifth power that's

two times two times two times two times

two five times

two to the fifth power is thirty two

now

if we plug in two

will we get the same answer or will we

get something different

2 squared plus 2 raised to the half 2

squared is 4

and 2 to the half is the square root of

2.

the square root of 2 has a decimal value

of about 1.4 so this is equal to 5.4

which is not equivalent to 32. therefore

number one

is not true

or it's not equivalent to x

raised to the fifth five over two power

so therefore e well not e but b is the

right answer

only statements 2 and 3 are true

19

if c is equal to 4 raised to the x

where x and c are both integers which of

the following expressions is equivalent

to 16 raised to the x plus 4 raised to

the x plus 2 power

so

how can we do this

so if we look at our answer choices

everything is in terms of c so somehow

we need to exchange x

for c

so we need to do some algebra here let's

start with uh the expression 16 raised

to the x plus 4

raised to the x plus 2.

now we can rewrite 16 as

4 squared

in order to convert x into c

we need the base 4.

so let's convert 16 into base 4. so 16

is

4 squared

now

last time we went over

this property of exponents we said x

squared times x cubed is x to the fifth

power

which is the same as x raised to the 3

plus 2.

so we want to do is we want to take an

expression in this form

and separate it into an expression that

looks like what we have on the left

so therefore if x raised to the 3 plus 2

power is equal to x squared times x

cubed

then

this expression 4

raised to the x plus 2 power is the same

as 4 to the x

times 4 squared because when you

multiply

common bases you can add the exponents

so

x plus two

is the same as or four basically x plus

two is the same as four to the x times

four squared

we can add x and two to get x plus two

if we go backwards

so you need to understand

that property of exponents in order to

uh

go from this step to this step that we

have here

so now 4 raised to the 2x is the same as

or 4 squared raised to the x is the same

as 4 to 2 x whenever you raise one

exponent to another you can multiply

the two exponents

and 4 squared is 16 so what we have is

16

times 4 to the x

now

instead of writing this as 4 to the 2x

i want to write it as

4 raised to the x squared

notice that these two expressions are

equivalent

2 times x is the same as x times 2

so 4 squared raised to the x is the same

as 4 raised to the x squared

now the reason why i chose to write it

that way is because at this point we can

now replace 4

raised to the x with c

so this is equal to c squared

plus 16 times c

so now we're going to factor out c

c is the gcf so if we take out c c

squared divided by c

is c and 16c divided by c is 16.

so we get c times c plus 16. so

therefore

answer choice a

is the correct answer for this problem

number 20

if the equations above are true

which of the following is a possible

value of y minus x

so

let's solve for x

if the absolute value of x plus two is

equal to seven

what is the value of x whenever you have

an absolute value equation

you can write two equations from it the

first equation is x plus two

is equal to positive seven and the

second equation is x plus two is equal

to negative seven the reason why we can

do that is because the absolute value of

positive seven is positive seven and the

absolute value of negative seven

is also positive seven and so that's why

we can separate into two equations

so for the equation on the left if we

subtract both sides by two

x is equal to positive five

and for the second equation if we

subtract both sides by two x is equal to

negative nine negative seven minus 2 is

negative 9.

now

for the other equation the absolute

value of y minus 3 is equal to 4. now

let's write two equations y minus 3 is

equal to positive 4

and y minus 3 is equal to negative 4.

so

if we add 3 to both sides 4 plus 3 is 7

and

negative 4 plus 3 is negative 1.

so these are the possible values

of x and y

so we need to see

what combination will give us one of the

answer choices that are listed here

so y minus x

starting with 7 if we choose 7 for y

we can subtract 7 by

the x value of 5

which is 2.

so 2 is not listed as an answer

we can also take the y value of seven

and subtracted by the x value of

negative nine

seven minus negative nine

is positive sixteen

and that answer is not listed here

so now we've used up y so now let's try

the other y value so if we use negative

one as y we can use positive five

for x

so negative one minus five is negative

six

that answer is not there

so now if we use negative one for y and

then the other x value negative nine

negative one minus negative nine is the

same as um

negative one plus nine

which is equal to eight

now that answer is listed there

so eight is a possible value of y minus

x

uh using these two equations

so as you can see there are four

possible values

negative 6

2

positive 16 and 8 but only 8 was listed

as one of our answer choices so

therefore 8 is the answer that we're

looking for so e is the right answer

21

if 4c plus b minus a over 7 is

equivalent to a

what is b in terms of a and c

so

how can we do this how can we find b in

terms of a and c

so what we need to do

is we need to solve for b

that's basically what the question

is

asking us to do

if we can isolate b on one side then a

and c will be on the other side of the

equation

so let's start with the expression

4c plus

b minus a over 7

equals a

now the first thing i would like to do

is get rid of the fraction so i'm going

to multiply

both sides by 7

so i'm going to multiply every term by

seven

four c times seven

is twenty eight c

and b minus a over seven times seven the

seventh will cancel

and then we'll have just b minus a left

over

and then a times 7 is 7a

so now i'm going to add a to both sides

so at this point the a's cancel on the

left

so what we now have is 28c

plus b

is equal to 8a

so now let's subtract both sides by

28c

so then b is equivalent to 8a

minus 28c

so now at this point we need to factor

the gcf

what is the greatest common factor

between 8 and 28

the greatest common factor is 4

4 can go into 8 and 28. so if we take

out

a 4

8 a divided by 4 is 2a

and negative 28c divided by 4 is

negative 7c

so therefore

we can see that a

is the right answer to this problem

22

if f of x is equal to two x plus five

and g of x is equal to the absolute

value of three minus x plus two

what is the value of g of f of one

so

here we want to evaluate a composite

function that's when one function is

inside another function

so f is inside of g

so first let's find out what f of one is

equivalent to

so that means we need to plug in one

into this equation

so let's replace x with one so two times

one plus five

is equal to two plus five which is seven

so therefore f of one is equal to seven

which means that we can replace f of one

with seven so we're looking for g of

seven at this point

so g of seven

is equal to the absolute value of three

minus seven plus two

three minus seven is equal to negative

four and the absolute value of negative

four is positive four so four plus two

is six

so therefore

g of f of one is equivalent to six so c

is the right answer for this problem

twenty three

if f of x is equal to the square root of

three minus x for all values where x is

equal to or less than one

and f of x is equal to five minus x

squared for all values where x

is greater than one what is the sum of f

of negative one and f of five

so what we really have is a piecewise

function

the function f of x can be broken into

two pieces

it can equal

the square root of three minus x and it

can equal five minus x squared

depending on the value of x

so

the first equation is true when x is

equal to or less than one

and the second equation should be used

when x is greater than one

so the goal for this problem is to find

the sum

of these two function values

so let's start with f of negative one

if we want to find out the value of f of

negative one should we use

the first equation or the second

equation

so

when is x equal to negative one

in this interval or in this interval

we know that it has to be true for the

first one because x is less than or

equal to negative one

so let's plug in negative one into the

first equation so it's three minus

negative one

which is the same as three plus one

and that's four and the square root of

four is equal to two

so now what about f of 5

should we use the first equation or the

second equation

5 is greater than 1 so

it's in the second equation

so it's going to be 5 minus

5 squared

if we replace 5 for x

so 5 squared is 25

and 5 minus 25 therefore is negative 20.

so now we can find a value of f of

negative one plus f of five because

we're looking for the sum of these two

function values

f of negative one we know it to be two

and f of five is negative twenty

so two plus negative twenty is equal to

negative eighteen and so therefore b

is the right answer for this problem

twenty four

let f of x comma y

be equal to y squared minus 5x

so if f of x comma 3 is equivalent to

negative 21 what is the value of x

so we have a function that contains two

variables x and y

and so this is equal to y squared minus

five x

and now we know that f of x comma three

is equal to negative 21.

so notice that the value of y is

equivalent to three

so we know that y is equal to three

so notice that the left side is equal to

the left side of the second equation so

therefore

the right side of the first equation

must equal the right side of the second

equation

so we're going to set those two equal to

each other so y squared minus 5x

is equal to negative 21 and we know that

y is three so this will allow us to

solve for x

three squared is equal to nine

and

we're gonna subtract

both sides by nine

so negative twenty one

minus nine

is equal to negative thirty

and so we're going to divide both sides

by

negative 5.

so therefore

negative 30 divided by negative 5

that is equal to positive 6

and so

d is the right answer for this problem

because that's all we're looking for we

just want to know what is the value of x

25

let the function f of c comma d

be equivalent to d squared plus c d

minus c squared

if f

3 comma e is equal to 61 and e is a

positive integer what is the value of e

so let's start with

the first equation that we have

the function is equal to d squared

plus c d

minus c squared

and we know that f

three comma e

is equal to uh 61.

now notice that c

is equivalent to three

and notice that d

is equivalent to e

and also

the right side of the first equation is

equal to the right side of the second

equation so we can therefore make the

statement that d squared

plus c d

minus c squared is equal to 61.

and we can replace

d with e and c with 3. so instead of

writing d squared we're going to write e

squared

plus

and then we're going to replace c for 3

so 3

times instead of writing d we're going

to write e again and then minus c

squared or 3 squared

and that's equal to 61.

so 3 squared

which is 3 times 3 that's equal to 9.

so notice that we have a quadratic

equation

e squared and e to the first power so we

might be able to factor it if not we can

complete the square or use the quadratic

equation

but usually these types of problems are

factorable

because the quadratic equation takes too

long

and if you're taking the sat you have to

do everything fast

so let's subtract both sides by 61.

so negative 9

minus 61

is equivalent to negative seventy

so let's see if this expression is

factorable

so what two numbers multiply to negative

seventy but add to positive three

so let's make a list

we have negative one and seventy

negative 2 and 35

3 doesn't go into 70

and 4 doesn't go into either but 5 goes

into it 14 times

so negative 5 and positive 14

7 goes into it so negative 7 and 10 and

this this works

negative 7 times 10 is negative 70 but

negative 7 plus 10 is positive 3. so to

factor it it's going to be e minus 7

and e plus 10.

so let's make some more space

so therefore

we can write two equations e minus seven

is equal to zero and e plus ten is also

equal to zero which means e is equal to

positive seven and e is equal to

negative ten

but notice that we have two answer

choices negative ten and positive seven

which one do we pick

now if we go back to the question it

said that e is a positive integer

so we can't use the negative value so

therefore

7 is correct answer choice e is the

right answer e is equal to positive 7.

26

let the function

h be defined by h of x

is equivalent to 7x plus 25.

so if the square root of h of b over

four is equal to nine

what is the value of b

so let's start

with the inside part of h of b over four

notice that there's no multiple choice

answers to select so this is a free

response problem

because

typically you'll see some of those

questions on the sat

so to find h of b over 4 we need to

replace x with b over 4.

so therefore h of b over 4

is equivalent to this expression

so starting with this problem

we can replace

the

h of b over 4 with this expression

so what we now have is the square root

of 7 times b over 4

plus 25

is equal to 9. so to get rid of the

square root symbol we need to square

both sides

so now

7b over 4

plus 25

is equal to 9 squared and 9 times 9 is

81

so at this point

let's go ahead and subtract to both

sides

by

25

so 81 minus 25

is equal to

that should be about

56 but

let me make sure my math is correct and

yes it's 56

so now we can cross multiply

whenever you have two fractions

separated by an equal sign you can cross

multiply 7b times 1 is 7b

and 56 times four

that should be 224

so now let's divide both sides by seven

so 224 divided by seven

is equal to 32

and so

that's the answer b has a value of 32.

27

if f of x is equal to

this expression

what is the value of f of x minus x

so f of x minus x is equal to

the expression that

f of x is equal to that's seven x

plus five

over three

minus four x minus seven

over three

minus x

so keep in mind this portion

is equal to f of x

so that's all we did we replace f of x

with what it equals to

uh these two fractions that are

subtracted to each other so now let's

see if we could simplify

the expression on the right side

and let's just see what happens

so

7x plus 5 over 3 we can separate that

into two fractions that's the same as

seven x over three

plus five over three

and we can separate four x minus seven

into two fractions by the same time

we're gonna distribute this negative

sign

so it's gonna be negative four x over

three

and then negative times negative seven

that's going to be positive

seven over three and then minus x

so let's combine these two fractions

because they're like terms

seven over three minus four over three

is basically three x over three

and we can combine these two

five thirds plus seven thirds five plus

seven is twelve

so that's twelve over three

and then minus x

now

three x divided by three is simply x and

twelve divided by three is four

and then we have minus x

x minus x is zero so the final answer

therefore is four

so f of x minus x is equal to four

twenty eight

if five x

is equal to twelve y

and y over z

is equal to eight over nine

then x over z is equal to

so how can we do this problem

well

what we need to do is we need to

rearrange some variables

so we need an equation that has only x

and z

so we need to remove y out of the

equation

so in the first equation let's solve for

y

so if 5x is equal to 12y

if we divide both sides by 12

we're going to get an equation that

states that y is equal to 5x divided by

12.

now in the second equation

y divided by z

is equal to eight over nine

we can replace y

with five x over twelve but before i do

that i'm going to rewrite the equation

like this y over z is the same as y over

one

times one over z

which is eight over nine

so therefore y over one which is

basically y we can replace that with

five x over twelve

times one over z and that's equal to

eight over nine

it's always better to separate this

fraction into two fractions by

multiplication

rather than plugging this in directly if

you plug it in right now it's going to

look like this 5x over 12

divided by z and then you're going to

have to fix that fraction now you have a

complex fraction

so i wanted to avoid the formation of a

complex fraction and so what i did is i

separate this into two

fractions by multiplication and it makes

it so much easier

so now

let's continue with what we have