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I can say , then

In an equation of the form
, the z-coordinate of
the vertex is at , but
here, so (0,f(0)) is the vertex, and to find

The vertex is at (0,2)
Now I translate this to (x,f(x))





(x,f(x)) = (-2,2)
The line of symmetry is
The graph is a min at
Here is the plot:

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Think of this as
The question to ask is: How many radians will
get you from back to again?
The answer is and any multiple of ,
so 4, 8, 12, 16, etc.
Now I can restate the problem as:

The 1st part just brings me back to where I started, and I'm
left with
------------------------------------
I'll show you a tougher example:

Starting from , how many multiples of will
bring me back to ? The answer is,
and any multiple of , like 8,16,24, etc.

Ignore the because that is how many times you go
completely around and back to , so the answer is:

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Get in the form

subtract from both sides
(1)
and

subtract from both sides
(2)
These have different slopes, so they must be independent
Here's the plot:

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First get it into the form or

Add to both sides

Divide both sides by

Now, I'll plot this

The graph is all the points that are above this line, but
not including the line itself.
A good way to check is to pick a point that is below the
line and make sure that it does not satisfy the inequality.
Like (5,11)


not true
so, (5,11) is not part of the graph of the inequality

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= number of $10 bills
= number of $5 bills
= number of $1 bills
given:
(1)
(2)
(3)
There are 3 equations and 3 unknowns,
so it's solvable
Subtract (2) from (1)
(1)
(2)

Substitute (3) into the result




and, since


and
(2)



Andrew had 36 $1 bills
check answer:
(1)
(1)


OK

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Yes

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Let one of the lengths =
The other length =
given:




and

The lengths are 70 and 35

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(miles/hour)x(hour) + (miles/hour)x(hours) + miles/hour)x(hours)




They drove 170 miles

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Your answer is correct. You got

Now multiply top and bottom by
Notice that this doesn't change anything, since

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Find the midpoint of the x-coordinates, then the y-coordinates.

and

I get (2) (-1, 3/2) as the answer

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Hi,
I'll explain this problem with a more simple problem that shows
the exact same idea.

OK, what you want to find is . What is the given inequality telling you?
If
then true
-----------------
If
then true
-----------------
If
then not true
----------------
In fact, if is -1, -2, -3, -4, or any minus number more negative than ,
the inequality is not true. It is true for any number that is more positive than
That includes the negative fractions like -1/3, -1/4, -1/5, etc. and also the
positive fractions like 1/3, 1/4, etc. and then it is true for all the positive numbers,
1, 2, 3, 4, etc.
--------------
Now I still want to find . The normal way to do this with an equality is to
divide both sides by

plus divided by minus is minus and
minus divided by minus is plus, so

and now just write this going from right to left
this is the same thing
But what happened? I said the original inequality was true for all numbers
greater than , or, in other words, .
Now I divided both sides by , and the result is telling me just the opposite.
This leads to a general rule:
------------------------------------------------------
Whenever you divide both sides of an inequality by a minus quantity,
you must reverse the inequality sign when you are done.
Following this law:



Now rewriting this from right to left:

Notice I reversed the inequality sign, and it is now true.
----------------
So, to solve your problem:

(I reversed the inequality sign)

(I just rewrote it going right to left)
This is true- try any number that is greater than or equal to ,
and the inequality becomes false.
for example:


not true
----------------
I hope all this helps and isn't too confusing.

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Here's how I do it:

Take 1/2 of the coefficient of , square it, and add it
to both sides.



Take the square root of both sides


and, also
(the other square root of 1)

from these, I get and , so
The factors are

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This is all the points below horizontal line through the y-intercept at .
The extra condition is that the points on the line not included
If the equation was , then the points on the line
would also be included

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The rate is

The tax is $5,949.96

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Using the point-slope formula






answer
check:
does the line include the given point?
(-1,-8)



You can check the other point

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I am a two digit number with the sum of 13. I have a 9 in the tens place
The number must be 94, because 9 + 4 = 13
-----------------------------------
I am a two digit number with the sum of 7. I have a 0 in the ones place
The number is 70, since 7 + 0 = 7
---------------------------
I am an even number less than 50 with the sum of 9
It's not 45, since that is an odd number
It could be 36, which is even
It can't be 27, which is odd
It could be 18, which is even
------------------------
I am a three digit number with a sum of 27
The number must be 999
There is no larger 3-digit number, and
any smaller number adds up to less than 27,
like 989, 9 + 8 + 9 = 26

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You are converting pounds to dollars
pounds x dollars/pound = dollars

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The 1st integer is
The numbers go odd, even, odd, even, etc.
so, the next odd number is
and thr next odd integer is
The sum is answer
to check: suppose



and


OK

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Let = liters of 40% solution needed
given:
liters of acid in 20% solution
= liters of acid in 40% solution
------------------------------------------------






4 liters of 40% solution are needed
check answer:



OK

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850%
-------------------



11.76%

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I notice that if I say , then I can say

This is obviously factored to

Now I can put it in terms of

There are 2 factors, and they are both

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There are possible ways to
arrange math books
There are ways to arrange chem books

Also, the math books could be on the left, chem on the right
or the opposite way, so the answer is

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(0,-2) has the form (x,y), so


Just plug these values into the equation



Yes, the point (0,-2) is a solution

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Let = number of nickels
Let = number of dimes
given:

(in cents)
By substitution:




and


There are 11 nickels and 22 dimes
check answer:




OK

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Let = Mary's age now
Let = John's age now
given:
(1)
Next is an equation that compares their ages when they are both 3 yrs older
(2)
------------------------
(2)
(2)
(2)
and
(1)
Multiply both sides by and subtract
(2) from (1)
(1)
(2)

and, from(1):



Mary is 12 and John is 9
check answer:
(2)
(2)
(2)

OK
and, also
(1)
(1)

OK

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Let = time to drive yesterday
= today's time
= time he wants to drive tomorrow
The average time is:







The times are:
26 1/3 min, 42 1/3 min, and 21 1/3 min

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I think I would multiply the top equation by

(now add)





Substitute this in the top equation:




I'll check the 2nd equation:




OK
You can check 1st

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You are adding rate:
(1 tub filled w/cold faucet)/(min to fill tub) + (1 tub filled w/hot faucet)/ (min to fill tub) = (1 tub filled)/(x min with both faucets open)

Multiply both sides by


min

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This is a rate problem.
Add the rates to get the resulting rate.
Filling the sink in a (+) rate
Emptying the sink is a (-) rate
The best way to understand it is
to express solution in words.
(1 sink filled)/(minutes to fill sink) - (1 sink drained)/(minutes to drain sink) = (1 sink filled)/(minutes to fill sink w/drain and faucet open)

Multiply both sides by


It takes 12 min for the sink to overflow with faucet
and drain open

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Does it go through (-4,5)?



OK
You can check the other point

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If the person goes from 1 city to another,
you only care about distances traveled, and
not time spent while not traveling.
To get distances:
d1 =
d2 =
d3 =
Add d1 + d2 + d3