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multiply both sides by a to clear the denominator on the right
multiply both sides by 6 to clear the denominator on the left
expand the left side
subtract 12 from both sides
factor or use the quadratic equation. Here is a good URL to help check your factoring -->http://72.3.253.76:8080/webMathematica3/quickmath/page.jsp?s1=algebra&s2=factor&s3=basic
in order for a product to be 0, one or more terms in the product must be 0. So in our case, either or
So solving those two equations, says either or are solutions to the problem.
Substitute those values for a into the original equation to check your answers.

2)Use the same steps as above.
Clear right side denominator
Clear left side denominator
this time, expand both sides
collect like terms
simplfy
divide both sides by 6

simplify
Here is a great website for math. You can use it to verify your work. For this problem the URL is --> http://www.wolframalpha.com/input/?i=5%2F(3x%2B1)+%3D+3%2F(3x-1)

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You have the correct x intercepts. Now you need to find the minimum point. You can see by looking at the function that the value will cause f(x) to be as small as it can get.
So calculate f(0)
=
Plot the points (0, -2), (0.534, 0) and (-0.534, 0)
Now draw a nice parabola thru them. If you need more points, just pick some x values (like 1 and -1), evaluate f(x) at those values for x and then plot those points too

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Without graphing, to find the minimum value, you need to look at the terms and see which one(s) can change. You know that is a constant, so that one won't change. You can see that the first term involves a factor of that includes a square. You know that the minimum a squared value can be is 0 (A squared value is always going to be positive or 0).
So the minimum value occurs when the first term is 0.
So what value of x makes

When x = -12.68, then

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Hard to say how many were served - we can say the minimum needed to make 63 peices.
We know that 63 halves were eaten, so 63/2 = 31.5 full pies.
She must have served at least 32 pies

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rule of thumb: the maximum area for a given rectangle perimeter is always a square. So in that case both the length and width are the same. Call it x

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Close ... but not quite
For each decrease of 13 miles (-13), the weight increases by 22 (22)

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mutliply both sides by
divide both sides by

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solve for y by subtracting 2x from each side. This isolates y by itself on the left hand side

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I think the question here is what does the contractor expect when a large enough set of similar deals happens.
You are told that contractor expects to make 20,000 70% of the time and lose 3000 30% of the time.
So what can the contractor expect if he/she does a bunch of these deals?

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This is a really bad question.
I assume what they are really asking is "How many ways can 72 be expressed as a product of two-digit even Integers". That answer would be:
None.
For a quick check, use the smallest 2 digit even number --> 10
If 10 is used, the 'other number' would be 7.2.
Any even number larger than 10 would make the other number smaller than 7.2. Since 7.2 is NOT a two digit integer, then the answer is "none".
So what makes this a bad question?
The fact that the problem asked for two digit even 'numbers'. Since 10 and 7.2 both contain two digits, and are both even, then they work. Since there is going to be a "really big' number of similar two-digit (albeit non-integral) numbers, makes the answer to this one a mess.

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Let the number be represented by x. Using the info in the problem:

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first figure out how many blue marbles there are



After you draw one red marble, there are 36 marbles left in the bag.
So the probability of picking a blue marble = = =

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18)
for y to be negative here, any x that is positive will do. Even x=0 would make y negative


You could solve it algebraically this way. In order for y to be negative, it must be "less than 0"


so any x that is larger than -2.5 will give you a y that is negative. Look at the plot and you can see the same thing

19) y= -4
This is a horizontal line. So all x, no matter what value has a y = -4

20)

In this much like 18.



Any x less than 2.5 yields a y that is negative

21)

Same idea here




Any x less than 1/6 works

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In order for a point to be an 'x-intercept', the y coordinate must be 0.
Let's substitute in y=0 and then attempt to solve for x



The Log(0) is undefined, so y cannot be 0.
You can also try it another way
As x increases, the value of approaches 0. However, 0 is the asymptote --> y never 'quite gets to 0, just really really close'

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Area = (x^2 + 6x + 8)
Width = (x +2)
Area = Length * Width
Area/Width = Length



If x=3 what are the area, length and width
Width = (x+2) = 3+2 = 5
Length = (x+4) = (3+4) = 7
Area = x^2 +6x + 8 = 3^2 + 6*3 + 8 = 9 + 18 + 8 = 35
Area = Length*Widht = 7 * 5 = 35

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You can see that at t=3, the term (t-3) above would evaluate to 0.
Since 0* anything = 0, then h(3) = 0.

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You are given roots, so you can figure out the quadratic. I will do the first one, you can use the same process to do the second one
1) roots are x=-8 and x=4
You know that at the roots, the value of the function is 0
You also know that for any product to equal zero, one or more terms in that function must be 0.
Thus
x =-8
x+8 = 0
x=
4
x-4=0
Now you have the two terms for your equation



This is the equation for the first problem. Since is it q quadratic, you know it is a parabola that opens up (opens up since the x^2 term is positive). You also know that the 'turning point' will be halfway between the two root. Since the roots are 12 units apart, the 'turning point' will be 6 units from each root.
-8+6 = -2
4-6 = -2
So find the value of f(x) when x=-2 and you'll have the minimum.




So the 'turning point is the point (2,-20)
Plot the funtion here --> http://www.wolframalpha.com/input/?i=x^2+%2B+4x+-+32+

Now do the same process for the other one. I am sure you can do it. If you still have questions, drop me an email

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1)





2)







check your answers

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The mean is the average. There are 4 values to include in the calculation






Check your answer.

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Each sheet is 1/2 inch thick

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plot is here --> http://www.wolframalpha.com/input/?i=x%2F4%2B10%3E2

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So that means
or
or
or

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You are given the density of silver and told about a silver sphere that has a mass of 0.295g.
So you need to find the diameter of a sphere that with that mass.
First let's figure out what volume the sphere must be.


You are asked to find the sphere's diamter in mm. So let's convert the volume to (mm)^3
There are 10 mm/cm, so the converion from (mm)^3/(cm)^3 = 10*10*10 = 1000(mm)^3/(cm)^3

You are ready ro solve


Now just grind out the value for r.
Once you have that, you need to remember you were asked for the diameter. So just double the radius and you are there.

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First find the linear scale.
You are told two areas are scaled at 1:2500
You know area is measured in squared linear dimension.
SO, what is the sqaureroot of 2500?

Thus the linear scale is 1:50
Now, you are asked about 'capacity of the petrol tanks'. That is a volume. And volume are in cubic dimensions.
So the scale needs to be cubed


wow

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The given equation is a line. So you need to find two points and then plot them.
For me, the easiest points to calculate are ones that have a 0 in one of the coordinates.
So let's find the point with where x=0



So one point is (0,3)
Now find the point with a y coordinate of 0



Another point is (4,0)
Now, plot those two points (0,3) and (4,0).
Then draw the line that contains those two points.
See this URL for a plot --> http://www.wolframalpha.com/input/?i=3x+%2B+4y+%3D12