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let d=days

to find the day that they have equal amounts ___ 5(22-d) = 10(18-d)

distributing ___ 110 - 5d = 180 - 10d

adding 10x and subtracting 110 ___ 5d = 70

dividing by 5 ___ d = 14

after the 14th day, Dan will have more than Mike

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"p=36= 2 (w x 2)+ 2 x w" ___ this is OK

"36= 2w x 4 + 2w" ___ this is not ___ the parentheses caused a problem??
___ distributive law not needed for multiplication __ just remove parentheses and multiply

36 = 4w + 2w ___ 36 = 6w ___ 6 = w

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let t = number of teens

"they split the $600 total cost equally" ___ 600 / t

" If they had had 5 fewer people, each person would have paid $10 more" ___ 600/(t - 5) = (600/t) + 10

multiplying by t(t-5) ___ 600t = 600t - 3000 + 10t^2 - 50t

subtracting 600t and dividing by 10 ___ 0 = t^2 - 5t - 300

factoring ___ 0 = (t + 15)(t - 20)

t + 15 = 0 ___ t = -15 ___ negative number not realistic

t - 20 = 0 ___ t = 20

substituting ___ $600 / 20 = $30

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try to maximize the number of less expensive books

5 goes into 86, 17 times with a remainder of 1

increasing the remainder in steps of 5, you need the smallest remainder divisible by 7 ___ like 21

5 * 13 = 65 and 7 * 3 = 21 ___ so 16 books

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factoring ___ (3x^2 - 5)(3x^2 - 2) = 0

3x^2 - 5 = 0 ___ 3x^2 = 5 ___ x^2 = 5/3 ___ x = ± sqrt(5/3)

3x^2 - 2 = 0 ___ 3x^2 = 2 ___ x^2 = 2/3 ___ x = ± sqrt(2/3)

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to go from R to M, it is -11 in the x direction and +14 in the y direction

since M is the midpoint, going from M to S uses the same displacements as going from R to M

S ((-7 -11),(8 +14)) ___ S (-18, 22)

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parallel means the same slope

so the x and y coefficients (A and B) are the same

substitute the coordinates of the point into the equation to find a new value for the constant term (C)

3(5) + 2(-2) + C = 0 ___ 15 - 4 + C = 0 ___ C = -11

the new equation is ___ 3x + 2y - 11 = 0

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google "roman numerals"

the wikipedia article is quite thorough

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dividing by 3 seems like a good idea ___ z > 13

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in the general form ___ A, B, and C are capitalized because they are whole numbers

you may have to multiply by a constant to "get rid of" fractions

also, by convention, A is always positive __ you may have to multiply through by -1 to accomplish this

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50% profit means that he sells the book for $30 ($20 cost + $10 profit)

40% discount means that the $30 is 60% of the price ___ $30 = (60%) p

$30 / .60 = $50 = p

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subtracting NS ___ S - NS = 5NT - TS

factoring ___ S - NS = T (5N - S)

dividing by (5N - S) ___ (S - NS) / (5N - S) = T

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(100%)(x) + (25%)(16) = (40%)(16 + x)

x + 4 = 6.4 + .4x

.6x = 2.4

x = 4

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factors of 100 that sum to 20 ___ like 10 and 10

(T + 10)(T + 10)

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186 = 16(3^2) + 3v

186 = 144 + 3v

42 = 3v

14 = v

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let x = yearly candy sales

"385 million $ ,which represented 7% of totla candy sales for the year"
___ 385M = 7% of x ___ 385M = (.07)x

dividing by .07 ___ 5500M = x

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multiplying by x ___ x + 1 = 2x - 2

adding 2 and subtracting x ___ 3 = x

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multiplying by x^2 ___ 7x^2 = - 12x - 2

adding 12x + 2 ___ 7x^2 + 12x + 2 = 0

the equation is not factorable, so you can solve it with the quadratic formula

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this equation is NOT factorable

solve it by using the quadratic formula



for a quadratic to be factorable, the discriminant (b^2 - 4ac) must be a perfect square

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the side length of a FINISHED square is 5 3/4 ___ but that is after you have turned over a 5/8 edge on EACH side

the starting side length must be 5 3/4 + 5/8 + 5/8 or 7 inches


b. one square is 7 x 7 and you need 48 ___ 7 x 7 x 48 = ?


c. the fabric is wide enough for 5 WHOLE squares ___ 36 / 7 = 5 1/7
___ you need 48 squares ___ there are 5 in a row, so you need 10 rows ___ because 9 is not enough 9 x 5 = 45
___ 10 rows of 7 inch squares is 70 inches ___ 10 x 7 = 70


d. the fabric is one inch wider than the 5 squares per row
___ so there is a strip of "extra" material one inch wide and 70 inches long ___ 1 x 70 = 70 square inches
___ also, only 3 squares are cut from the 10th row ___ this leaves 2 squares of "extra" material ___ 2 x 7 x 7 = 98 square inches
___ so the square inches of left over fabric is ___ 70 + 98 = ?

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(16z-9) + (4z+3) = 90

20z - 6 = 90 ___ 20z = 96 ___ z = 96 / 20 = 4.8

substitute the value of z to find the angle measures

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your answer is correct and your technique is fine for integer (whole number) exponents

if you need to find fractional or decimal exponents, a scientific calculator has an exponentiation key

windows has a built-in calculator that can be set to scientific mode
___ start >> programs >> accessories >> calculator

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if r is a root (solution), then x-r is a factor

(x + 10)(x - 4) ___ x^2 + 6x - 40 = 0

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let x = ties

"won 12 more games than they tied" ___ so x+12 = wins

2(x+12) + 1(x) = 57

distributing ___ 2x + 24 + x = 57 ___ 3x = 33 ___ x = 11

so, 11 ties and 23 wins

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x=2 is a vertical line (y is not defined and so can have any value)

the perpendicular line must be horizontal and of the form y=?

this means A and B have the same y-coordinate (1)

x=2 is the bisector of AB so the distance from A to x=2 is the same as the distance from x=2 to B
___ 2 - -4 = ? - 2 ___ 6 = 8 - 2 ___ so the x-coordinate of B is 8

point B is (8,1)

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multiplying by 2+x ___ -x-2 < 2-x ≤ 2+x

adding x ___ -2 < 2 ≤ 2x+2 ___ you can drop the first term because there it has no effect on the x

subtracting 2 ___ 0 ≤ 2x

dividing by 2 ___ 0 ≤ x

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r - c = 2

r + c = 3

adding equations ___ 2r = 5

dividing by 2 ___ r = 2.5

substituting ___ 2.5 + c = 3 ___ c = .5

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x = 5

y = -5

by Pythagoras, r = sqrt(50) or 5 sqrt(2)

sec = r / x ___ or sqrt(2)

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let x=shorter leg

by Pythagoras ___ x^2 + (2x + 8)^2 = (3x - 8)^2

x^2 + 4x^2 + 32x + 64 = 9x^2 - 48x + 64

0 = 4x^2 - 80x = x^2 - 20x = x (x - 20)

x = 0 ___ not a realistic solution

x - 20 = 0 ___ x = 20

longer leg ___ 2(20) + 8 ___ 48

hypot ___ 3(20) - 8 ___ 52

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two comments

first ___ the number is not irrational, it is 45/13 (the ratio of two whole numbers)

second ___ x is the distance one-way ___ the total distance is 90/13

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distributing ___ 9 - 3x > 8x - 4

adding 3x + 4 ___ 13 > 11x

dividing by 11 ___ 13/11 > x