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1)die = singular, dice = plural
2) The probability that a die, thrown once, will show 2 AND 3 is 0.
The probability that a die, thrown once, will show 2 OR 3 is 2/6, or 1/3.
Thus the expected number of times a 2 or a 3 will show is (1/3)*90 =30 times.

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-3/2 - 4 < -2m < 7/2 - 4
==> -11/2 < -2m < -1/2
==> 11/4 > m > 1/4, or (1/4, 11/4) in interval notation.

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Sir/Madam,
The basis of all graphing exercises, from the simplest (plotting points) to the the most complicated (e.g., algebraic geometry), is an appreciation of the basic Cartesian coordinate system. Give your son lots of exercises on plotting random points on the Cartesian system. Then ask him if he can form geometrical objects, any geometrical object, by connecting dots (points) on the coordinate system. After doing this for some time (probably after having had enough of this simple exercise), introduce order and structure by illustrating how to form a straight line using two points (, ) and (, ). Always keep in mind the position of the line with respect to the origin (0,0), and whether the line falls or rises from left to right. Don't be discouraged if he doesn't get it right away: for many people it takes time to understand the most basic of mathematical concepts (which might actually include me.) =)

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<==>
<==> ==>
The critical numbers of the inequality are 0 and 3.
At the interval (, 0), .
At the interval (0,3), .
At the interval (3, ), .
The critical numbers 0 and 3 don't satisfy the inequality.
Therefore the solution set is (, 0)U(3, ).

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Without replacement:
==> rwb = 24 = 2*3*4.
Since r+w+b = 9, there 6 possibilities for the ordered triple(r,w,b):
(2,3,4)
(2,4,3)
(3,2,4)
(3,4,2)
(4,3,2)
(4,2,3)

With replacement:
==> , which is impossible, because r,w, b are positive integers satisfying r+ w + b = 9

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12*11*10 = 1,320 ways

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Consistent dependent, meaning infinitely many solutions.

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Since { ,, , ..., } are linearly independent, then the equation + ...+ has only the trivial solution = ...= . (is the zero vector, and 's are scalars.)
We have to assume that none of the 's are equal to zero, otherwise,
{ ,, , ..., } would contain the zero vector, automatically making the set linearly dependent. Hence
+ ...+
==> for every n = 1,2,3,...,k, and { ,, , ..., } is a linearly independent set.

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4w - 3v = 4(-i-6j) - 3(-3i + 7j) = -4i - 24j + 9i - 21j = 5i - 45j

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Area = square inch.
Diagonal = inch.

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: Defects are independent of manufacturer location
: Defects are associated with manufacturer location
p = 0.05, with degrees of freedom = df = (2 - 1)(4 - 1) = 3.
Critical value is 7.82.

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<==>
<==>
==> {x: }U {x:} = {x:}

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==> ==> x = 21, the total number of gallons Pablo's tank can hold.

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9:00 am.

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(a)
==> .
(b)

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15m - 5(m+6) > -6m + 30,
15m - 5m - 30 > -6m + 30,
10m - 30 > -6m + 30,
16m > 60,
m > 60/16 = 15/4,
or {m| m > 15/4}.

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Let and , where
is in the 4th quadrant.
Then

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If the function is , then f(-6) exists, and is equal to .

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==>
==>
==>
<==> 0 = (x + 2)(x+ 5) ==> x = -2, -5.
Both values will never make the denominator 0, so they are the final answers.

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Let r = speed at the 1st part of trip ==> time for 1st part = 48/r
==> r - 5 = speed at 2nd part of trip ==> time for 2nd part = 19/(r-5).
both from the formula t = d/r = distance/rate.
Then
<==> 48(r-5) + 19r = 3r(r-5)
<==>
<==> <==> (r - 24)(3r - 10) = 0
==> r = 24, 10/3
Eliminate r = 10/3, because it would make r - 5 negative.
Hence the speed in the 1st part is 24 mph, and the speed in the 2nd part is 19 mph.

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If your rational function is , then the domain is obtained as follows:
From the expression , the critical numbers are 2, 3, and 6. These critical numbers partition the real number line into the intervals (, 2), (2, 3), (3, 6), (6, ).
At (, 2), by using the test point x = 0.
At (2, 3), by using the test point x = 2.5.
At (3, 6), by using the test point x = 4.
At (6, ), by using the test point x = 7.
The critical numbers 2 and 6 are not included in the domain, as these will make the denominator equal to 0. We want those intervals that will make , because the whole expression is under the square root symbol. The critical number x = 3 is included in the domain.
Hence the domain of the rational function is (2,3]U(6, ).

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Cosine Law:
==>
==> ==> c = 11.981, to 3 decimal places.

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Measure of steepness of a line = Rise/Run = for two points (,) and (,) on the line.

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The line must have the equation 2x + 5y = c. To find c, use the coordinates of the point (-3,4):
2*-3 + 5*4 = -6 + 20 = 14
Therefore the equation of the line is 2x + 5y = 14.

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2 triangles are possible.
From the sine law, , we get to 5 decimal places.
==> 58'36"
1st possibility:
A =
B = 58'36"
C = 1'24"
2nd possibility:
A =
B = - 58'36" = 1'24"
C = 58'36"
Just get the sides using several applications of the sine law.

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miles.

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5.6*0.583 =3.2648 grams of pure gold.