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If you plot , you get:





From the graph, we see that the lowest point is (1.5,-5). So the smallest value in the range is -5 (since this is the smallest y value that is on the graph).


This means that the range is which looks like [) in interval notation.

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Since "y varies inversely as x", this means that . Because "y=14 when x=8", this means that we can plug those values in to get: . Multiply both sides by 8 to isolate k to get: . So the constant k is


So the inverse-variation equation is


To find the y values when x=10, 15, and 20, simply plug these values into 'x' of the equation and simplify.

Ex: When x=10, then . So if , then


I'll let you do the rest.

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Start with the given system of equations:





Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

Start with the first equation


Subtract from both sides


Rearrange the equation


Divide both sides by


Break up the fraction


Reduce



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

Since , we can now replace each in the second equation with to solve for



Plug in into the second equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute to


Multiply


Multiply both sides by the LCM of 5. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



Distribute and multiply the LCM to each side



Combine like terms on the left side


Add 18 to both sides


Combine like terms on the right side


Divide both sides by -7 to isolate x



Divide





-----------------First Answer------------------------------


So the first part of our answer is:









Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms and reduce. (note: if you need help with fractions, check out this solver)



-----------------Second Answer------------------------------


So the second part of our answer is:









-----------------Summary------------------------------

So our answers are:

and

which form the point








Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of (red) and (green) and the intersection of the lines (blue circle).

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a) Just pick a word that has a length of 1. So any letter will do.


b) Since each word has a specific length (and only ONE specific length), this means that the relation L is indeed a function. If L mapped say the word 'the' to the numbers 3 AND 4 (say hypothetically that 'the' was 3 characters and 4 characters long), then L wouldn't be a function since that input yielded two outputs.


So in general, if every input into the relation yields only one output, then that relation is a function.

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I'll do the first two to get you started.

# 1



Looking at we can see that the first term is and the last term is where the coefficients are 1 and -63 respectively.

Now multiply the first coefficient 1 and the last coefficient -63 to get -63. Now what two numbers multiply to -63 and add to the middle coefficient -2? Let's list all of the factors of -63:



Factors of -63:
1,3,7,9,21,63

-1,-3,-7,-9,-21,-63 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -63
(1)*(-63)
(3)*(-21)
(7)*(-9)
(-1)*(63)
(-3)*(21)
(-7)*(9)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2

First Number	Second Number	Sum
1	-63	1+(-63)=-62
3	-21	3+(-21)=-18
7	-9	7+(-9)=-2
-1	63	-1+63=62
-3	21	-3+21=18
-7	9	-7+9=2

From this list we can see that 7 and -9 add up to -2 and multiply to -63


Now looking at the expression , replace with (notice combines back to . So it is equivalent to )




Now let's factor by grouping:


Group like terms


Factor out the GCF of out of the first group. Factor out the GCF of out of the second group


Since we have a common term of , we can combine like terms

So factors to


So this also means that factors to (since is equivalent to )



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



Answer:
So factors to


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

# 2



Looking at we can see that the first term is and the last term is where the coefficients are 1 and -24 respectively.

Now multiply the first coefficient 1 and the last coefficient -24 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient 2? Let's list all of the factors of -24:



Factors of -24:
1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -24
(1)*(-24)
(2)*(-12)
(3)*(-8)
(4)*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2

First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

From this list we can see that -4 and 6 add up to 2 and multiply to -24


Now looking at the expression , replace with (notice combines back to . So it is equivalent to )




Now let's factor by grouping:


Group like terms


Factor out the GCF of out of the first group. Factor out the GCF of out of the second group


Since we have a common term of , we can combine like terms

So factors to


So this also means that factors to (since is equivalent to )



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



Answer:
So factors to


The rest of the problems are all similar to these two. So I'll let you finish up.

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Start with the given equation.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the solution is

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Start with the given equation.


Distribute.


Combine like terms on the right side.


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.


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

Answer:

So the solution is

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I'm assuming you want to factor this right?



Start with the given expression


Group like terms


Factor out the GCF out of the first group. Factor out the GCF out of the second group


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


So factors to


In other words,

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Since , you are correct in saying that you are going to take the square root of each number. One way you can do it is to write 16 as and 100 as and then realize that (where . So this means that .

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Assuming that the letters can only be used once, you have 7 letters for the first slot, 6 letters for the second (since one has been used already), 5 for the third, etc. until you have one left over for the last slot. Multiply these values to get: 7*6*5*4*3*2*1 = 5040


So there are 5,040 different seven-letter 'words'. Note: a 'word' in this case is just a string of letters (it may or may not be a real word)

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I think you made a typo. The problem should be "Show that 2x^2-12x+23 is equal to 2(x-3)^2+5"


Start with the given expression.


Factor out the coefficient . This step is very important: the coefficient must be equal to 1.


Take half of the coefficient to get . In other words, .


Now square to get . In other words,


Now add and subtract inside the parenthesis. Make sure to place this after the "x" term. Notice how . So the expression is not changed.


Group the first three terms.


Factor to get .


Combine like terms.


Distribute.


Multiply.


So after completing the square, transforms to . So for all real values of 'x'.

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Hint: use the law of cosines where 'a', 'b', and 'c' are the lengths of the triangle and 'A' is the angle opposite the side 'a'. So this problem will make and you'll let 'b' and 'c' be equal to the remaining values given. Let me know if this helps at all.

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Start with the given equation.


Distribute.


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

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Start with the given equation.


Multiply both sides by 'y'.


Add 'y' to both sides.


Factor out the GCF 'y' from the right side.


Divide both sides by to isolate 'y'.


So the solution is


Here's the confirmation of the answer.

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Start with the given equation.


Multiply both sides by 4.


Distribute.


Get every term to the left 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 .


Take the square root of to get .


or Break up the expression.


or Break up the fraction for each case.


or Reduce.


or Simplify.


So the solutions are or

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Start with the given equation.


Distribute.


Combine like terms on the left side.


Add to 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.


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

Answer:

So the solution is

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It's not stated, but there is another fact about this figure that HK = HK. Since IK = JH and HI = KJ also, we have 3 pairs of congruent sides which means that SSS will be used in this case.

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Since you're given that AB = CD, AD = CB, and BD = BD, this means that you are basically showing that these three pairs of sides are congruent to one another. So this would then mean that the SSS (Side Side Side) Property of Congruence would be used.

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Since the equation is in slope-intercept form , where 'm' is the slope and 'b' is the y-intercept, this means that the slope is .


Similarly, for , this equation is also in slope-intercept form and the slope is as well. So the two slopes are equal.


What this means is that the life expectancies are both increasing at the same rate.


To find out which years men have a greater life expectancy than women, simply set M(x) (life expectancy of men) greater than W(x) (life expectancy of women) like this





Plug in and


Multiply both sides by 1000 to clear out the decimals.


Distribute and multiply.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Simplify.


Since this inequality is NEVER true for any value of 'x', this means that the original inequality is also NEVER true.


Since there are no solutions to the original inequality, this means that the life expectancy of men will never exceed the life expectancy of women.


Note: I would make sure that you copied the problem down correctly (or ask if there are any typos).

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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.


Multiply 3 and 2 to get 6.

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


Factor out the GCF


Combine like terms.


So simplifies to .


In other words,

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Take note that . So . Also, keep in mind that


Start with the given equation.


Square the individual terms (see above).


Combine like terms (this just amounts to adding fractions).


Multiply both sides by 4.


Divide both sides by 5.


Cancel out the common terms.


Simplify.


Take the square root of both sides.


or Break up the "plus/minus" to form two equations.


or Take the square root of to get .


or Combine like terms.


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


Answer:


So the solutions are or .

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The trick here lies in simplifying

Start with the given expression


Factor out a negative 1


Break up the square roots using the identity


Replace with (remember )


Now lets simplify :


The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. When you take the square root of this perfect square, you will get a rational number.


So let's list the factors of 32


Factors:
1, 2, 4, 8, 16, 32


Notice how 16 is the largest perfect square, so lets factor 32 into 16*2


Factor 32 into 16*2



Break up the square roots using the identity



Take the square root of the perfect square 16 to get 4



This means that the expression simplifies to


So the expression simplifies to (just reintroduce back in)


In other words,


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

Going back to , this expression simplifies to . From there, it reduces to


So the solutions are or

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Start with the given equation.


Distribute.


Combine like terms on the left side.


Add to both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the solution is

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Check out this solution.

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Start with the given equation.


Subtract 2L from both sides.


Divide both sides by 2 to isolate W.


So the answer is

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We basically have this triangle set up:





To find the unknown length, we need to use the Pythagorean Theorem.


Remember, the Pythagorean Theorem is where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.


Since the legs are and this means that and


Also, since the hypotenuse is , this means that .


Start with the Pythagorean theorem.


Plug in , ,


Square to get .


Square to get .


Combine like terms.


Rearrange the equation.


Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).


Simplify the square root.


Approximate the right side with a calculator.


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


Answer:


So the solution is approximately which means that the hypotenuse is roughly 15.6 (rounded to the nearest tenth) units long.

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Remember that you CANNOT divide by zero. So set the denominator equal to zero to get . Solve for 'w' to get . So when , the denominator is equal to zero which is what we don't want. This means that we must make the restriction that for the expression to make sense.

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Simply set up the equation where 'x' is the number of miles she runs in 72 minutes. Notice how the ratio of 4 to 30 is equal to the ratio of x to 72.


Start with the given ratio


Multiply both sides by 72


Multiply


Reduce.


So our answer is which in decimal form is which means that she can run 9.6 miles in 72 minutes.

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First, let's expand

Start with the given expression.


Rewrite as


Expand. Recall that


Distribute.


Multiply.


Now combine like terms.


So expands to .


In other words, .


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


This means that then becomes


Now let's finish up...



Start with the given expression.


Expand.


Distribute.


Multiply.


Now combine like terms.


So expands to .


In other words, .



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


Answer:


So fully expands out to


In other words,



Note: if this problem was more complicated than it should have been, then maybe there's a typo somewhere.

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The degree of any monomial is just the sum of all of the exponents. So the degree of is and the degree of is . The degrees of 7 and are 0 and 5 respectively.


The degree of the entire polynomial is the largest degree of all of the monomials present. So the degree of is 15 since this is the largest degree among the individual monomials.

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Start with the given equation.


Plug in


Simplify


Divide both sides by -6.


Simplify


So when , giving us the point which is the x-intercept.