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7 days in a week
set the two costs equal to find the "break even" milage

7(63) + .18m = 7(126) + .09m

subtracting [.09m + 7(63)] ___ .09m = 7(63)

dividing by .09 ___ m = 7(700) = 4900 miles

you need to drive MORE THAN this to save money with rate 2

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3 = 5 e^(-.04t)

.6 = e^(-.04t)

ln(.6) / -.04 = t

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parents stick together...

each term contains 2p
___ factoring this gives ___ 2p (q - 2p + r)

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"The height of a cylinder is twice the diameter" ___ h = 2d ___ h/2 = d ___ h/4 = r

A = LA + ends = [2 * pi * r * h] + [2 * pi * r^2] = [2 * pi * h/4 * h] + [2 * pi * (h/4)^2]

A = [2 * pi * (h^2 / 4)] (5 / 4) = 5/8 (pi * h^2)

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77/88 - 16/88 = (2x)/88

77 - 16 = 2x

61 = 2x

61/2 = x ___ this is as reduced as it gets

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the x-intercept(s) (when y=0) are the roots ___ find with the quadratic formula

the y-intercept (when x=0) is just the c term (ax^2 + bx + c)

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distributing ___ -8x + 24 + 12x = 15 - 2x - 3

collecting terms ___ 4x + 24 = -2x + 12

adding 2x and subtracting 24 ___ 6x = -12

dividing by 6 ___ x = -2

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the table is of x and y pairs that make the equation true
you pick a value for x, and then find the y value

if x is 0, then y must be 1

if x is -4, then y must be 5

and so on...

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there are two foci, in a VERTICAL line, symmetric about the x-axis

if this were a hyperbola, there would be NO x-intercepts

so you have an ellipse
___ centered at (3,0)
___ the length of the minor axis is ___ 2b = 7 - -1 = 8 ___ so, b = 4
___ distance from center to focus (c) is 2
___ a^2 = b^2 + c^2 = 16 + 4 = 20

[(x - 3)^2 / 16] + [(y - 0)^2 / 20] = 1

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the lines should have equations

solve the equations for y
___ solutions of the form, y = mx + b

the x-coefficient (m), is the slope of the line

the constant term (b), is the y-intercept (where the line crosses the y axis)

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"A store owener sells two pairs of running shoes for each pair of high tops" ___ r = 2h

"He plans to order 144 pairs of shoes" ___ r + h = 144

two unknowns (r and h) and two equations

substituting ___ (2h) + h = 144 ___ h = 48

substituting ___ r = 2(48) = 96

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2f + 4s = $5.90

not enough information to solve
___ in general, you need as many independent equations as there are unknowns

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extract (factor out) any WHOLE roots

√100 √5 + √100 √3

10 √5 + 10 √3

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let t="tens digit" and u="units digit"

"The value of a two-digit number is twice as large as the sum of its digits" ___ 10t + u = 2(t + u)
___ 10t + u = 2t + 2u
___ 8t = u ___ (there is only one pair of digits that works here)

"If the digits were reversed, the resulting number would be 9 less than 5 times the original number"
___ 10u + t = 5(10t + u) - 9
___ 10u + t = 50t + 5u - 9
___ 5u + 9 = 49t

substitute the value of u from the 1st part to find the value of t
___ then substitute t back to the 1st part to find the value of u

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you've got the times and speeds reversed

they overtake when they have travelled the same DISTANCE (rate x time)

the shorter time goes with the faster speed

60 t = 48 (t + 1/3) ___ the Smiths had a 20 min head start

12 t = 16 ___ t = 4/3 or 1:20

8:20 + 1:20 = 9:40

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(6s - 7) (s + 6)

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99 > 3x > 1000

33 > x > 333 1/3 ___ 34 to 333


999 > 4x > 10000

249 3/4 > x > 2500 ___ 250 to 2500


250 to 333 ___ 84 integers

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5x + 2 > 2 ___ and ___ 5x + 2 < -2

5x > 0 ___ x > 0

5x < -4 ___ x < -4/5

x > 0 ___ and ___ x < -4/5

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let x=sunflower, so 3x=raisins

(4)(x) + (1.5)(3x) = 34

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dividing by 12000 ___ (1 + r)^2 = 9 / 4

taking sqrt ___ 1 + r = 3 / 2

subtracting 1 ___ r = 1 / 2 ___ or .5 ___ or 50%

pretty good investment...

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(3x + 14) (x - 2) = 0

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(t-5)^2 + 2(t-5) = 0

(t-5) [(t-5) + 2] = 0

(t-5) (t-3) = 0

t-5 = 0 ___ t = 5

t-3 = 0 ___ t = 3

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(y + 1) (y - 7) (y + 3)

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difference of squares

[(y - 5) + 6q] [(y - 5) - 6q]

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[(a + 4) - 1]^2 or (a + 3)^2

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dividing by sec^2 ___ tan^2 + 1 = sec^2

multiplying by cos^2 ___ sin^2 + cos^2 = 1

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let x="grams of 50%", so 200-x="grams of 75%"

(50%)(x) + (75%)(200 - x) = (66%)(200)

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let x="grams of 50%", so 200-x="grams of 75%"

(50%)(x) + (75%)(200 - x) = (66%)(200)

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let x="pounds of $.75 seed", so 50-x="pounds of $1.10 seed"

x(.75) + (50 - x)(1.10) = 50(.90)

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2(x + 9)(x - 1)

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(x + 14)(x - 10)