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1/2*4m -3m -5/2,,,,,,,check problem to assure we copied correctly
.
(1/2 * 4m) -3m -5/2
.
2m -3m -5/2
.
-1m -5/2
.
-m-2.5
.

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Let x be larger of two variables, y the other
.
(1) x+y=11,,,,,or x=11-y
.
(2) x/y = 2 +Rem 2 = 2 + 2/y
.
subst (1) into (2)
.
(11-y) /y = 2 + 2/y
.
multiply thru by y
.
11-y = 2y +2
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11-2=2y+y
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9=3y
.
3=y,,,,,,,,,x=11-y=11-3=8
.
.
CHECK,,,(1) 8+3=11,,,,ok
.
(2) 8/3=2 +2/3,,,,,ok

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Domain is the set of the independent variable, usually x,,
.
while Range is the set of the dependent variable, usually y
.
Physically Domain might be time while Range is Weight, for a diet log
.
or Domain might be elapsed time vs Range , distance, for a travel log,
.
Usually Domain and Range are expressed as, "all Real Numbers"
.
or " all positive Real Numbers"
.
Exceptions occur when the value of the function, dependent variable, is
.
UNDEFINED,(denominator is zero),or COMPLEX (square root of a negative number)
.

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Let P = positive number,,,and, N= sum of negative numbers
.
Given,,P+N>0
.
P>-N, that is,,Positive Number greater than sum of negative numbers
.
Examples, P=5,,N=4=1+1+1+1, or 5-1-1-1-1>0
.
or,,,,,P=5, N=3=1+1+1,,,,or 5-1-1-1>0
.
or,,,,,,P=5, N=2=1+1,,,,,or 5-1-1 >0

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Area = Length * Width
.
(x^2 +50x) = L*x
.
x(x+50) = L*x
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(x+50) = L
.
check...A=LW,,,A=(x+50)(x) = x^2 +50x,,,,ok

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(A) Cost = .10*Area + .25* Perimeter
.
If l=30,,& w=20,,,,Area = L*W,,,,Per = 2L + 2W
.
cost = .10(30*20) +.25( 2*30+2*20)
.
Cost = .10(600) + .25(100)
.
Cost = 60 + 25 = $ 85
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(B) Let L=40,,,,find W
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Cost = .1A + .25P = .1(40*W) + .25( 2*40+2*W)= 4W + 20 + .5W = 4.5 W +20
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If Cost is $ 85,,85=4.5W +20,,,,65 = 4.5 W,,,,W= 14.444 in.
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(C),,,, Cost = .1Area + .25Per
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C= .1(L*W) + .25(2L+2W)
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C= .1LW + .5L +.5W
.
(D) C-.5W = L (.1W +.5)
.
(C-.5W) / (.1W +.5) = L,,,,,,checking with original(20*30=$85),,,ok

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3/x +1/4 = 5/8
.
3/x = 5/8 - 1/4
.
3/x = 5/8 -2/8
.
3/x = 3/8
.
x=8
.
check,,,3/8 +1/4 =3/8 +2/8 =5/8,,,,,ok

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x^(3/2) = 27
.
raising both sides of eqn to (2/3)
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x^1 = 27^(2/3) = {27^(1/3)}^2 = 3^2 = 9
.
or x=9
.
check,,,9^(3/2) = 27,,,,ok

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Area of Rectangle = Length * Width
.
but L = W +3
.
A = L*W = (W+3)* W= W^2 + 3W
.

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Perimeter of rectangle = 2* Length + 2* Width
.
P =250 = 2L +2W,,,,,dividing by 2
.
125 = L+W,,,,,W=125-L
.
Area = L*W = L*(125-L) = 125L -L^2
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making a table
.
L,,,,,,,Area
.
1,,,,,,,124
.
60,,,,,,3900
.
62,,,,,,3906
.
62.5,,,,3906.25,,,,,max,,,,L=W=62.5
.
63,,,,,,3906
.
The max area of the rectangle occurs when L=W=62.5 ft,,,,
.
This makes rectangle into a square
.

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Please see problem # 205691

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Let (s+q) =a
.
or substituting,,,a^2 -64
.
factoring as difference of squares
.
(a+8)(a-8)
.
substituting back original variables
.
{ (s+q) +8}{(s+q)-8}
.
{s+q+8}{s+q-8}
.

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The sphere, liquid, and radius, makes a right triangle, with a hypotenuse equal to the spherical radius of 18 cm, and one leg of length 9 cm.
.
Using the pythagorean theorem, the radius of the liquid can be found.
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c^2 = a^2 +b^2,,,,or 18^2 = 9^2 + r^2,,,,r^2 = 18^2 -9^2 = 243,,,,r=15.588 cm
.
or the diameter = 2*15.588 = 31.177 cm
.

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1st integer,,,n,,,,,,,,,11
2nd integer,,,n+2,,,,,,,13
3rd integer,,,n+4,,,,,,,15,,,,,sum to 75,,,,,avg is 75/5 = 15
4th integer,,,n+6,,,,,,,17
5th integer,,,n+8,,,,,,,19
.
6th integer,,,n+10,,,,,,21
7th integer,,,n+12,,,,,,23
8th integer,,,n+14,,,,,,25,,,,,,sum to 125,,,,,avg is 125/5=25
9th integer,,,n+16,,,,,,27
10th integer,,,n+18,,,,,,29
.
TOTAL,,,,,,,,,10n +90,,,,,,,,,,,,,,,,,,,,,,,,,,,,,sum to 200
.
If (10n +90)/10 =20,,,,,n+9 =20,,,,n=11
.
substitute back in above

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(1) 3n^n +3n^q =9,,,,n^n = (9-3n^q)/3 = 3-n^q
.
(2) 2n^n +6n^q =10
.
subst (1) into (2)
.
2(3-n^q) +6n^q =10
.
6-2n^q +6n^q =10
.
4n^q=4
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n^q=1,,,,,,therefore q=0,,,,anything raised to 0 =1,,,
.
subst into (1),,,,3n^n +3n^0 =9
.
n^n +n^0 =3
.
n^n +1 =3
,
n^n =2
.
n= 1.56
.
check(1) 3*1.56^1.56 +3*1.56^0 =9
.
6 +3=9,,,,,ok
.
(2) 2*1.56^1.56 = 10 - 6*1.56^0
.
4= 10 -6 =4,,,,ok
.

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5 % of X = 2.5,,,,, "%" means per centum (or per hundred),,"of" means multiply
.
5/100 * X=2.5
.
X= 2.5 / (5/100) = 2.5 / .05 = 50
.
check,,,,2.5 / 50 = .05 = 5 %,,,,ok
.

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(3)y= kx,,,,,,,that is y varies directly with x
.
8 = k * (16),,,,,,,when y=8, x=16,,,,,,find k
.
k=8/16 =1/2
.
Variation is,,,,y = 1/2 * x
.
when x= 46,,,,,,,,y = (1/2) * 46 = 23
.
check,,,8/16 = 23/46,,,,,ok
.
(5) y=kx,,y varies directly with x
.
2=k*4,,,,when y=2,,,x=4,,,,find k
.
k=1/2
.
Variation is ,,,,y= 1/2 *x
.
y= (1/2)*45 =22.5
.
check 22.5 /45 = 2/4,,,,ok
.
(7) y= k/x,,,,,,,y varies in directly with x
.
10=k/25, y=10 when x=25,,,find k
.
250 = k
.
Variation is,,,,y = 250/x
.
y= 250/35 = 7.14,,,,,find y if x=35
.
check,,,y=k/x,,,xy =k = 10 * 25 = 7.14 * 35 = 250,,,ok
.

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{ ( x+9) / (x-3) } * {(x^3 -27) / (x^2 +3x+9)}
.
factor cubic,,,x^3 -27 = x^3 -3^3 = (x-3)(x^2 +3x +9)
.
{(x+9)/(x-3)} * { (x-3)(x^2 +3x +9)/ (x^2 +3x+9)}
.
simplify by canceling like terms
.
(x+9)
.
check,,,let x=2,,, original and answer = 11,,,ok

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f(x) =2^x,,,,,on reflection about y axis,,f(x) =2^(-x)
.
{ logic,,,f(x) =y,,,, reflection about y axis,,,has same y value ,,but
.
negative x value
.
plot graph to check
.

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sqrt44 = sqrt(4 *11) = sqrt4 * sqrt(11) = 2sqrt(11)
.
sqrt(11/16) = sqrt(11) / sqrt(16) = sqrt(11) / 4 = (1/4) sqrt(11)
.
check,,,sqrt(44) = 6.63 = 2 sqrt(11),,,,ok
.
sqrt(11/16) = .829 = (1/4)sqrt(11),,,ok
.

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-3x-3y=5,,,,,remember slope intercept form ,,,y=mx+b
.
-3y=3x+5,,,,,divide thru by (-3)
.
y= (-1)x +(-5/3),,,,,,,,,line of slope(-1),,y intercept of -(5/3)
.
a parallel line has same slope, different y intercept,
.
such as,,,,y= (-1)x +(+5/3)
.
but this in standard form is,,,3y = -3x + 5,,,or,,,-3x -3y = -5
.
to generalize, a set of parallel lines ,,,has same form as original
.
except change constant.
.

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log(4) {2x+1} = log(4){x-3} + log(4){x+5}
.
log (2x+1) / log(4) = log(x-3)/log(4) + log(x+5) / log(4)
.
multiply thru by log(4)
.
log (2x+1) = log(x-3) + log (x+5)
.
log(2x+1) = log(x-3)(x+5),,,,,,loga +logb = log ab
.
2x+1 = (x-3)(x+5) = x^2 +2x -15
.
0 = x^2 -16,,,,factor
.
0 = (x+4)(x-4),,,,set equal to zero
.
x= 4, -4
.
check,,,(+4),,, log(4){9) = log(4){1)+log(4){9),,,but ,,,log(1)=0,,,,ok
.
(-4),,,log(4){-7} = log(4){-7}+log(4){1},,,but,,log(4){1}=0,,,,ok
.

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looking at a typical hundreds chart,
.
in each set of 0 to 10,,(1-10,,11-20,,21-30,,etc) we have one 8, and one nine, for a total of two.
.
this is times 8, for sets of 10 up to 80.
.
then the 80's and 90's, give 10 each.
.
for typical 100 set, 8*2 +10 +10 = 36, 8"s or 9"s
.
this happens for 8,,,,100 sets,,(0-100,100-200, ....700-800),,,or 8*36 = 288
.
all,,800's,,have an 8,,,or +100
.
and all 900's have a 9,,,or +100
.
This brings total to 288 +100+100=488,,,8"s or 9"s,,,up to 1000
.
total without is 1000 - 488 = 512,,,or answer (c)

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(E n )/9 =40
.
{E (n(1) +1) + n(2) +2 +......... } / 9 = ?
.
(E n)/9 + { 1+2+3+4+5+6+7+8+9 } /9 = ?
.
(En)/9 + { 45/9} = ?
.
40 +5 = ?
.
45 = ?

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Remember that,,,a^3 -b^3 = (a-b)(a^2 +ab+b^2)
.
64y^3 -125z^3
.
4^3y^3 -5^3z^3
.
(4y)^3 -(5Z)^3,,,,,or a=(4y),,,b= (5z)
.
(4y-5z){ (4y)^2 + (4y)(5z) +(5z)^2 }
.
(4y -5z)(16y^2 +20yz + 25 z^2)
.
or answer (c)

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3x^5 +3x^3 = 3x^3 ( x^2 +1) = 3x^3 ( x+i)(x-i)
.
Remember that, i^2 = -1
.

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180
.
2----90
.
,,,,,2---45
.
,,,,,,,,,5----9
.
,,,,,,,,,,,,,,3-3
.
summary, 2*2*3*3*5 = 180,,,,ok
.
or 2^2, 3^2, 5,,,,,,which is answer (a)

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(-4,-2),,,(1,1)
.
m= (y2-y1)/(x2-x1) = (1-(-2))/(1-(-4)) = 3/5 ,,,,slope is positive 3/5
.
a rough graph, showing the points, connecting line, and rise / run confirms the calculation.

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2x -5y = -12
.
if x= -1,,,,,y= ?
.
2(-1) -5y = -12
.
-2 -5y = -12
.
-5y = -10
.
y=2,,,,,,,,(-1,2) satisfies eqn
.
check,,2(-1) -5(2) = -2 -10 =-12 ,,,,,,ok
.

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net gain per day is (3-2) =1 m / day
.
to gain 12 m,, would take 12 days,,,(12 m/1 m/day = 12 days)
.
.
Recognize that on the 10th day, the snail will start at the 9 m level,,and
.
gain 3 m during the day to just reach 12 m.
.
If some sort of platform exists to prevent his sliding back,
.
he could achieve his journey at the end of 10 days .
.

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a+bi,,,,,,is an expression of a complex number
.
Real part (a) ,,,,complex part (b)
.
(5/3) + (1) i,,,,,,,shows (a) = (5/3) ,,,,,(b) =(1)
.
.
it is really a +/- bi,,,,,so just recognize real and complex in the
.
same way for a-bi as we did above for a+bi
.
.
Usually this is plotted on an x - y axis graph,
.
with x as the real part(a) and y as the imaginary part (b)