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Start with the given expression.


Factor to get .


Factor to get .


Factor to get .


Factor to get .


Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words,

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Start with the given expression.


Factor to get .


Factor to get .


Factor to get .


Factor to get .


Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


Simplify.


Distribute


So simplifies to .


In other words,

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Start with the given expression.


Multiply the first fraction by the reciprocal of the second fraction .


Factor to get .


Factor to get .


Factor to get .


Factor to get .


Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


Simplify.



So simplifies to .


In other words,

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Start with the given equation.


Get everything to one side


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "x":


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


or Break up the expression.


So the solutions are or


which approximate to or

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Start with the given expression.


Factor into


Factor into


Break up the square root using the identity .


Take the square root of to get .


Take the square root of to get .


Rearrange and combine the terms.

==================================================

Answer:


So simplifies to


In other words, where .

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Check out this previous solution.

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Simply add up the numbers and divide that sum by 7 (since there are 7 values)




So the answer is C)


Note: normally the mean is represented in decimal form.

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Start with the given equation.


Factor the last denominator


Multiply EVERY term by the LCD to clear out the fractions.


Simplify


Distribute


Multiply


Subtract 1 from both sides.


Combine like terms.


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "x":


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the possible solutions are or


However, recall that , this means that the only solution is

=============================================================

Answer:

So the solution is

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I'll do the first three to get you going in the right direction.


# 1

Start with the given expression.


Multiply both the numerator and denominator by


Combine the roots.


Multiply


Take the square root of 25 to get 5



So

---

# 2


Start with the given expression.


Factor 24 to get 4*6


Cancel out the common terms.


Simplify


Combine the roots


Divide


Simplify


So where

---

# 3


Start with the given expression.


Factor 15 into 5*3


Cancel out the common terms.


Simplify


Combine the roots.


Factor 8 into 4*2


Cancel out the common terms.


Simplify


Divide the variable terms by subtracting the exponents.


Subtract


Factor into note: there are 6 terms.


Break up the square root using the identity .


Take the square root of to get .


Take the square root of to get .


Multiply.

==================================================

Answer:


So simplifies to


In other words, where



Edit: The other poster is correct in saying that "x" doesn't need to be non-negative, BUT recall that . If we assume that (ie it is NOT negative), then which simplifies things greatly. If "x" can be any number, then you must include the absolute value.

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Just list all the multiples of 6 and take off any numbers that are multiples of 10. So here are all the numbers from 1 to 200 that are are divisible by 6 and not by 10:


6, 12, 18, 24, 36, 42, 48, 54, 66, 72, 78, 84, 96, 102, 108, 114, 126, 132, 138, 144, 156, 162, 168, 174, 186, 192, 198


There are 27 numbers here.

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Start with the given compound inequality.


Multiply EVERY term by the LCD to clear out any fractions.


Simplify


Multiply


Add to all sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide all sides by 21.


Reduce.


So the solution is

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I'm assuming that the problem is right???


Recall that if , then




In your case, , which means that and


Now derive f and g to get: and



Now plug these functions into the formula to get:





Now multiply every term by the inner LCD x+2 and simplify






you could perform more simplifications, but that would only change how the answer looks.

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A)

Note: is the first point R and is the second point S .


Start with the slope formula.


Plug in , , , and


Subtract from to get


Subtract from to get


So the slope of the line that goes through the points R and S is


-------------------------------------------------------------

B)

Now simply flip the fraction to get . Finally, change the sign to get . So the slope of ANY perpendicular line to segment RS is

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Let w = speed of the wind (in mph)


First, let's set up the equation involving the trip against the wind:


Start with the distance-rate-time formula


Plug in and (the wind slows him down, so you must subtract "w" from 18)


Divide both sides by .


Rearrange the equation


-----------------------------------


Go back to the distance-rate-time formula


Plug in and (the wind speeds him up, so you must add "w" to 18)


Divide both sides by .


Rearrange the equation

-------------------------------------


Now recall that he "can bicycle 8 mi against the wind in the same time it takes to bicycle 12 miles with the wind". So this means that the times "t" are the same variable. This means that we can plug one equation into another:


Start with the given equation.


Plug in


Cross multiply


Distribute.


Subtract from both sides.


Add to both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


Divide (either use a calculator or long division)


----------------------------------------------------------------------

Answer:

So the solution is which means that the speed of the wind is 3.6 mph.


Note: if you are expecting a whole number for an answer, I would verify that you have the correct problem written down.

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Let x = first integer


So x+1 is the next integer, x+2 is the next integer after that, etc...


So because "The number 9 can be written as the sum of 9 consecutive integers", this means that


Start with the given equation.


Remove the parenthesis.


Combine like terms on the left side.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the solution is which means that the first integer is -3


So the 9 consecutive integers are:

-3, -2, -1, 0, 1, 2, 3, 4, 5


If you multiply them together, you'll get 0 (since 0 is being multiplied here).

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Start with the given expression


Simplify to get . Note: If you need help with simplifying square roots, check out this solver.


Simplify to get .


Since we have the common term , we can combine like terms


Factor out the GCF


Combine like terms.


So simplifies to .


In other words,


You can verify this with a calculator.

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Start with the given equation.


Distribute


Subtract "a" from both sides.


Subtract 1 from both sides.


Combine like terms.


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "a":


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the solutions are or

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a)

Start with the given function.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So the wrench is 80 feet in the air after 1 second.


-----------------------------------------------------------------------

b)

Start with the given function.


Plug in (ie make the height equal to zero).


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "t":


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the possible solutions are or

But remember that a negative time value doesn't make much sense. So we'll ignore


So the only solution is which means that it takes 2 seconds for the wrench to hit the ground.

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There isn't a rule, but it's conventional to place the variable first like so . Why? Well later you'll learn about polynomials and you'll find that the constant terms are placed last.


Note: the expression is the exact same expression (ie for all values of "m").

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Start with the given expression.


Factor out the GCF 25 from the first radicand


Factor out the GCF 9 from the second radicand


Break up the roots using the identity


Take the square root of 25 to get 5


Take the square root of 9 to get 3


Factor out the GCF


Combine like terms.



So where

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Start with the given expression.


Rewrite 8 as


Rewrite as


Rewrite as


Break up the roots using the identity


Evaluate the cube root of to get 2


Evaluate the cube root of to get "y"


Multiply


Factor out the from the first two terms. note: this step isn't necessary, but your book does it.


=======================================================
Answer:


So simplifies to



In other words,

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Let's simplify this expression using synthetic division


Start with the given expression

First lets find our test zero:

Set the denominator equal to zero

Solve for x.

so our test zero is -2


Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.(note: remember if a polynomial goes from to there is a zero coefficient for . This is simply because really looks like

-2	|	1	-2	0	8
	|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)

-2	|	1	-2	0	8
	|
		1

Multiply -2 by 1 and place the product (which is -2) right underneath the second coefficient (which is -2)

-2	|	1	-2	0	8
	|		-2
		1

Add -2 and -2 to get -4. Place the sum right underneath -2.

-2	|	1	-2	0	8
	|		-2
		1	-4

Multiply -2 by -4 and place the product (which is 8) right underneath the third coefficient (which is 0)

-2	|	1	-2	0	8
	|		-2	8
		1	-4

Add 8 and 0 to get 8. Place the sum right underneath 8.

-2	|	1	-2	0	8
	|		-2	8
		1	-4	8

Multiply -2 by 8 and place the product (which is -16) right underneath the fourth coefficient (which is 8)

-2	|	1	-2	0	8
	|		-2	8	-16
		1	-4	8

Add -16 and 8 to get -8. Place the sum right underneath -16.

-2	|	1	-2	0	8
	|		-2	8	-16
		1	-4	8	-8

Since the last column adds to -8, we have a remainder of -8. This means is not a factor of
Now lets look at the bottom row of coefficients:

The first 3 coefficients (1,-4,8) form the quotient



and the last coefficient -8, is the remainder, which is placed over like this





Putting this altogether, we get:



So

which looks like this in remainder form:

remainder -8

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Let r=Robert's age right now


So if "in x years from now, Robert will be k years old", this means that we simply add "x" to "r" to get "k" like so: . Now solve for "r" to get . So Robert's age right now (in terms of "k" and "x") is


Now just subtract off "m" more years from to get


So the answer is C) k-x-m


-----------------------------------------------------------------------


Note: if this doesn't make any sense, just plug in some values. Let's say that Robert is 12 years old. So . In two years, he will be 14 (which means k=14). So this means that we'll make . Now how old was Robert 5 years ago (note: m=5)


So plug in , and into to get:



So he was 7 years old 5 years ago (from his current age of 12)

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Let
a = # of adults that attended
c = # of children that attended


Since a total of 254 people attended, this means that . Also, because "The Jurassic Zoo charges $28.00 for each adult admission and $12.00 for each child" and "The total bill for 254 people from a school trip was $3864.00", we know that


So we have the system of equations:





In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for "a", we would have to eliminate "c" (or vice versa).


So lets eliminate "c". In order to do that, we need to have both "c" coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero and cancel out.

So to make the "c" coefficients equal but opposite, we need to multiply both "c" coefficients by some number to get them to an equal number. So if we wanted to get 1 and 12 to some equal number, we could try to get them to the LCM 12. However, we want one number equal to 12 and the other equal to -12.


Because one "c" coefficient is already equal to 12, we need to multiply both sides of the first equation by -12 (to get the "c" coefficient equal to -12).





Start with the first equation.


Multiply the both sides of the first equation by -12.


Distribute and multiply.


So we have the new system of equations:





Take note that the "c" coefficients are equal but opposite.


Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms.


Simplify.


Divide both sides by to isolate .


Reduce.


------------------------------------------------------------------


Now go back to the first equation.


Plug in .


Multiply.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


============================================================================

Answer:


So the solutions are and .


This means that 51 adults attended and 203 children attended.

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Here's one possible ordered pair: (0,6)


6(0)+6=6 ---> 0+6=6 ---> 6=6 ... which is true

------------------------------

And another possible ordered pair: (1, 0)

6(1)+0=6 ---> 6+0=6 ---> 6=6 ... which is true

------------------------------

And another possible ordered pair: (2, -6)

6(2)-6=6 ---> 12-6=6 ---> 6=6 ... which is true


------------------------------

And another possible ordered pair: (3, -12)

6(3)-12=6 ---> 18-12=6 ---> 6=6 ... which is true

------------------------------

etc...

So this shows us that there are an infinite number of ordered pairs that satisfy this equation.

Note: only specific pairs of numbers satisfy the equation. You can't just make up two random numbers and hope that they work.

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Start with the given expression.


Plug in and


Rewrite as


Square -5 to get 25


Multiply


Combine like terms.


Reduce


So when and


So either you entered the problem in wrong or there's a typo somewhere in your book. Is there something missing? Repost with the clarifications.

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Table of Contents:
Part i)
Part ii)
Part iii)
Part iv)


Given Matrices:













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Part i)


Start with the addition of the two given matrices


Add up the matrices by adding the corresponding entries


Add

===========================================

Answer:

So

---

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Part ii)

To find the inverse of the matrix , we can follow these steps:


Step 1) Find the determinant


The determinant of is . So this means that

note: since the determinant is NOT equal to zero, the matrix inverse exists.

--------------------------------
Step 2) Swap the values


Now switch the highlighted values to get


--------------------------------


Step 3) Change the sign


Now change the sign of the highlighted values to get


--------------------------------


Step 4) Multiply by the inverse of the determinant


Multiply by to get


Plug in to get


--------------------------------

Step 5) Perform scalar multiplication and simplify.


Multiply by EVERY element to get


Multiply to get


=================================================================

Answer:
So the inverse of is


This means that if then


note: to verify your work, multiply the matrices and and you should get the matrix (the identity matrix).

---

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Part iii)

Take note that the dimensions of matrices A and B are 2x3 and 2x3 respectively. Since the inner dimensions do NOT match, this means that the matrix product AB is NOT defined. In other words, AB does NOT exist.

note: if A and B are mxn and nxp matrices, then the product AB is defined. It is only defined if the number of columns of A equals the number of rows of B. Otherwise, AB does NOT exist.

===========================================

Answer:

So the matrix product AB is not defined (ie it doesn't exist).

---

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Part iv)

I'm sure you meant to write correct? If so, then...


Step 1: Transpose matrix B to get . You do this by converting each row into a column (and vice versa).




Step 2: Multiply matrices and C. Since the inner dimensions are equal, this means that the matrix product is defined.



Since the first matrix is a 3 by 2 matrix and the second matrix is a 2 by 2 matrix, this means that the resulting matrix will be a 3 by 2 matrix.

So the final resulting matrix will look like:





note: the "x"s are just placeholders for now



--------------------------------------------------




Multiply the corresponding entries from the 1st row of the first matrix by the 1st column of the second matrix. After multiplying, add the values:


1st row, 1st column:



So the element in the 1st row, 1st column of the resulting matrix is . Now let's update the matrix:


--------------------------------------------------




Multiply the corresponding entries from the 1st row of the first matrix by the 2nd column of the second matrix. After multiplying, add the values:


1st row, 2nd column:



So the element in the 1st row, 2nd column of the resulting matrix is . Now let's update the matrix:






================================================================================




Multiply the corresponding entries from the 2nd row of the first matrix by the 1st column of the second matrix. After multiplying, add the values:


2nd row, 1st column:



So the element in the 2nd row, 1st column of the resulting matrix is . Now let's update the matrix:


--------------------------------------------------




Multiply the corresponding entries from the 2nd row of the first matrix by the 2nd column of the second matrix. After multiplying, add the values:


2nd row, 2nd column:



So the element in the 2nd row, 2nd column of the resulting matrix is . Now let's update the matrix:






================================================================================




Multiply the corresponding entries from the 3rd row of the first matrix by the 1st column of the second matrix. After multiplying, add the values:


3rd row, 1st column:




So the element in the 3rd row, 1st column of the resulting matrix is . Now let's update the matrix:


--------------------------------------------------




Multiply the corresponding entries from the 3rd row of the first matrix by the 2nd column of the second matrix. After multiplying, add the values:


3rd row, 2nd column:




So the element in the 3rd row, 2nd column of the resulting matrix is . Now let's update the matrix:










==============================================================================


Answer:


So



In other words,




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Start with the given expression.


Rearrange the terms.


Multiply 3 and -4 to get -12


Combine the roots using the identity


Multiply 2 and 10 to get 20


Factor 20 to get 4*5. Note: one of the factors must be a perfect square.


Break up the roots using the identity


Take the square root of 4 to get 2.


Multiply -12 and 2 to get -24


So

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a)

Start with the given function.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So the wrench is 80 feet in the air after 1 second.


-----------------------------------------------------------------------

b)

Start with the given function.


Plug in (ie make the height equal to zero).


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "t":


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the possible solutions are or

But remember that a negative time value doesn't make much sense. So we'll ignore


So the only solution is which means that it takes 2 seconds for the wrench to hit the ground.

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Let L=length of garden and W=width of garden


Since the garden is "4m longer than it is wide", this tells us that

Now remember, the area of any rectangle is


Start with the area of a rectangle formula


Plug in (the given area) and


Rearrange the terms.


Distribute


Divide


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "W":


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the possible solutions are or


However, you can't have a negative width. So the only solution is

So the width of the garden is 8 m.


Now add 4 m to 8m to get 4+8=12 m. So the length is 12 m


======================================================================
Answer:


So the length and width of the garden are 12 meters and 8 meters.

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First let's find the slope of the line through the points and


Note: is the first point and is the second point .


Start with the slope formula.


Plug in , , , and


Subtract from to get


Subtract from to get


Reduce


So the slope of the line that goes through the points and is


Now let's use the point slope formula:


Start with the point slope formula


Plug in , , and


Rewrite as


Distribute


Multiply


Add 3 to both sides.


Combine like terms.


Remove the trailing zero


Simplify


So the equation that goes through the points and is


Notice how the graph of goes through the points and . So this visually verifies our answer.
Graph of through the points and