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statement II and III are equivalent.
statement II states:
TODAY IS NOT sunday AND TOMORROW IS NOT monday
statement III states:
IT IS NOT THE CASE THAT (today is sunday and tomorrow is monday).
THE CASE is the statement that (today is sunday and tomorrow is monday).
IT IS NOT THE CASE THAT negates the statement within the case.

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the following picture shows you what i am talking about.

label your trapezoid ABCD starting from top left then to top right then to bottom right then to bottom left.
since this is an isosceles trapezoid, sides AB and CD are parallel, and sides AD and BC are congruent.
you are showing the ratio of the sides as 4:5:10:5
AB is assigned to 4
BC is assigned to 5
CD is assigned to 10
AD is assigned to 5
since these are ratios, and not actual sizes, then multiply each of them by a common factor which we'll call x.
the sides of your trapezoid are then equal to:
AB is equal to 4x
BC is equal to 5x
CD is equal to 10x
AD is equal to 5x
x is a common ratio that will apply to any size trapezoid with the sides in the same ratio.
you know that the height of the trapezoid is 8 and the perimeter is 48.
you want to find the area.
we can use the perimeter to find the sides of the trapezoid.
we just add up all the lengths of the sides and make them equal to 48 and solve for x.
you get:
4x + 5x + 10x + 5x = 48
combine like terms to get:
24x = 48
divide both sides of this equation by 24 to get:
x = 2
the lengths of your sides are now:
AB is equal to 8
BC is equal to 10
CD is equal to 20
AD is equal to 10
if you now drop a perpendicular from A to intersect with CD at E, and drop a perpendicular from B to intersect with CD at F, then you can find the area of the isosceles trapezoid.
AB is equal to 8
CD is equal to 20
dropping the perpendiculars from A and B to intersect with E and F on CD divides CD into 3 segments.
DE is equal to 6 and EF is equal to 8 and CF is equal to 6 making a total of 20 for the overall segment of CD.
you have now formed a right triangle of ADE with a right angle at E.
you have also formed a right triangle of BCF with a right angle at F.
the vertical lines are the altitudes (height) of the trapezoid.
since you know that the height of the trapezoid is 8, then the height of these altitudes is also 8.
to find the area of the trapezoid, use the formula A = 1/2 * (b1 + b2) * h
A is the area.
b1 and b2 are the smaller and the larger bases of the trapezoid..
h is the height.
b1 is equal to AB which is equal to 8.
b2 is equal to CD which is equal to 20.
h is equal to either AE or BF which are each equal to 8.
using these figures, the formula becomes:
A = 1/2 * (8 + 20) * 8 which becomes:
A = 1/2 * (28) * 8 which becomes:
A = 14 * 8 which becomes:
A = 112 square units.

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the numbers under the radical sign operate with the same arithmetic rules as the same numbers outside of the radical sign.
the difference is that they are under the radical sign, and so the final answer will be whatever the root indicated by the radical sign.
example:
you have 1/2 + 3/4
you would combine this by putting everything under a common denominator that will make it become 2/4 + 3/4 = 5/4.
whether or not that expression is under a radical sign (making it a radicand) or outside a radical sign, the operations are the same.
the difference is that the answer is still under the radical sign.
example:
you have sqrt(1/2 + 3/4)
that becomes sqrt(5/4) after the arithmetic operations are performed on it.
so the answer to your question is that the arithmetic operations are the same, except, in one case they are performed outside the radical sign, and in the other case, they are performed inside the radical sign.
some terminology.
in the expression square root of (7), the square root symbol is the radical and the 7 is the radicand (the number under the radical symbol)
if you followed the logic above, you'll see that the operations on (1/2 + 3/4) were the same, regardless of whether those numbers were within the radical sign, or outside of it.
if this answers your question, then fine.
if not, then send me a specific problem you are working on where you decided to question whether the operations are the same.

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your equation is:
x/5 + 5y = m/n
you want to solve for y.
subtract x/5 from both sides of the equation to get:
5y = m/n - x/5
divide both sides of the equation by 5 to get:
y = (m/n)/5 - (x/5)/5
(m/n)/5 is equivalent to m/(5n)
(x/5)/5 is equivalent to x/25)
your equation becomes:
y = m/(5n) - x/25
you can put everything under a common denominator to get:
y = 25m/(25*5*n) - 5nx/(25*5*h) which becomes:
y = (25m - 5nx) / (125n)
you can factor this further to get:
y = 5*(5m-nx) / (125n)
you can simplify this further to get:
y = (5m-nx) / (25n)
i think this is as far as you can go.
to confirm you did the process correctly, give some numbers to the variables and see if the equations are the same.
i let:
m = 2
n = 3
x = 4
using these values, the original equation of:
x/5 + 5y = m/n becomes:
4/5 + 5y = 2/3
subtract 4/5 from both sides to get:
5y = 2/3 - 4/5
divide both sides by 5 to get:
y = (2/3)/5 - (4/5)/5 which becomes:
y = 2/15 - 4/25
i'll stop there and see if i can duplicate that with the final equation.
the final equation is:
y = (5m-nx) / (25n)
after substitution this becomes:
y = (10 - 12) / 75
i have 2 equations.
the final equations shows up as y = (10 - 12) / 75
the original equation shows up as y = 2/15 - 4/25
i will get the original equation to have a common denominator.
the original equation becomes:
y = 10/75 - 12/75 which becomes (10 - 12) / 75
both equations are identical so the transformation was performed correctly.

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if the equation winds up with an equality and no variables, then you are dealing with an infinite number of solutions.
example:
3 = 3
0 = 0
etc.
if the equation winds up with no equality and no variables, then you are dealing with no solutions.
example:
2 = 3
0 = 5
etc.
an example of a system of equations with infinite number of solutions.
x + y = 2
2x + 2y = 4
you solve this system of equations by multiplying the first equation by 2 to get:
2x + 2y = 4 (first equation multiplied by 2)
2x + 2y = 4 (second equation)
when you subtract the first equation from the second equation, you get:
0 + 0 = 0 which becomes 0 = 0
this indicates an infinite number of solutions.
any value for x and any value for y that satisfies one of the equations will automatically satisfy the other equation.
for example:
if x = 5 and y = -3, then x + y = 2 becomes 5 - 3 = 2 which becomes 2 = 2 which is good.
plugging those same values into the second equation gets:
2x + 2y = 4 becomes 2*5 - 2*3 = 4 which becomes 10 - 6 = 4 which becomes 4 = 4 which is good.
any combination of x and y that satisfies one of the equation will satisfy the other.
an example of no solutions is as follows:
x + y = 2
2x + 2y = 7
when you multiply the first equation by 2 to eliminate one of the variables, you wind up eliminating all of the variables and you get:
2x + 2y = 4 (first equation multiplied by 2)
2x + 2y = 7 (second equation)
when you subtract the first equation from the second equation, you get:
0 + 0 = 3 which becomes 0 = 3.
this is false, so there is no solution to this system of equations.
we can graph both the infinite number of solutions and the no solution to show you how the graph will look.
your first 2 equations were:
x + y = 2
2x + 2y = 4
solve for y in both equations and you will get:
y = -x + 2
y = -x + 2
these equations are identical and so their graphs will coincide and look like the same line.
your second 2 equations were:
x + y = 2
2x + 2y = 7
solve for y in both equations and you will get:
y = -x + 2
y = -x + 7/2
these equations have the same slope but have a different y intercept so they are parallel to each other. this means they will never intersect which means you have no common solution.
note that all equations are in the slope intercept form.
that form is y = mx + b
m is the slope and b is the y intercept.
if the slopes are the same and the y intercepts are different then the lines are parallel and will never intersect.
if the slopes are the same and the y intercepts are the same, then the lines are identical and you have an infinite number of solutions.
the graph of the first 2 equations where we had an infinite number of solutions is shown below:

the graph of the second 2 equations where we had no solution is shown below:

in the first graph, the 2 lines are superimposed on each other because the equations are identical so it looks like you have one line, but you really have 2.
the only way to know that is to remove one of the equations from the graph and then you will see that the graph is still there.

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an adjacent side of 3 and an opposite side of 7 leads to an angle of 66.80140949 degrees.
you cannot have an angle of 15 degrees with an opposite side of 7 and an adjacent side of 3.
i will assume your angle is 66.80140949 and solve for the hypotenuse.
fyi, the angle was derived as follows:
tangent (x) = opposite / adjacent = 7/3 = 2.3333333333333
arc tangent (2.333333333333) = 66.80140949 degrees.
you can use either sine or cosine formula to find the hypotenuse.
sine (66) = opposite / hypotenuse = 7 / hypotenuse
this leads to hypotenuse = 7 / sine(66) which equals 7.615773106.
cosine (66) = adjacent / hypotenuse = 3 / hypotenuse
this leads to hypotenuse = 3 / cosine (66) which equals 7.615773106
both methods lead to a hypotenuse of 7.615773106.
i do not know where you got the 15 degrees from.
the tangent of 15 degrees is equal to .267949192 which cannot be duplicated by 7/3 or 3/7.

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your original garden is 12 * 16 = 192 square feet.
the width is 12 and the length is 16.
you added 120 square feet to it.
the new garden is 192 + 120 = 312 square feet.
you added the same amount to the length and the width.
we'll call that x.
your formula for the new garden is:
(x + 12) * (x + 16) = 312
multiply out the factors to get:
x^2 + 28x + 192 = 312
we'll solve by completing the squares.
subtract 192 from both sides of this equation to get:
x^2 + 28x = 120
take 1/2 of 28 and square it and then add it to the right side of the equation.
you'll get:
x^2 + 28x = 120 + 196
this becomes:
x^2 + 28x = 316
take 1/2 of 28 and form the squaring factor of:
(x + 14)^2 = 316
that's your completing the squares equation.
take the square root of both sides of this equation to get:
x + 14 = +/- sqrt(316)
subtract 14 from both sides of this equation to get:
x = +/- sqrt(316) - 14
solve for x to get:
x = 3.776388835
or:
x = -31.77638883
since x can't be negative, you are left with:
x = 3.776388835
replace x in your original equation for the new garden with that number and you get:
(x + 12) * (x + 16) = 312 becomes:
(3.776388835 + 12) * (3.776388835 + 16) = 312
simplify that equation to get:
(15.776388835) * (19.776388835) = 312
multiply that out and you'll get:
312 = 312, confirming that the value for x is good.
the exact dimensions of the new garden are:
length = 19.776388835
width = 15.776388835
to convert this to feet and inches, multiply the fractional part by 12 and add that number of inches to the integral part, which is the feet.
you will get:
length = 19 feet 9.3166660915 inches
width = 15 feet 9.3166660915 inches
you can round to whatever decimal place on the inches that you want.
looking at your answer, it looks like you got one dimension ok, but you forgot to get the other dimension.
you got what i call the length.
you need to add the same amount you added to the length, to the width, to get 15 feet 9 inches.
you should have added 3.776388835 feet which became close to 3 feet 9 inches as you showed, if you are rounding to the nearest inch.

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click on the following link to see your picture in a separate window.
your picture is here
The secant is GB.
all the other ones are chords, except for BP and PC which are radii.
AD is both a chord and a diameter.
AP and PD are also radii.
FB, FC, BE, and EC are chords
a secant is a line that intersects a circle at 2 points.
if the line terminates at the boundaries of the circle, then it is a chord.
if it extends beyond the boundaries of the circle at one or both ends, then it is a secant.
if the line intersects the circle at one point only, then it is a tangent.
check the following reference for additional information.
http://www.mathopenref.com/circle.html

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there are 3 basic rules of exponentiation used to solve this.
the first one is:

the second one is:

the third one is:

you can apply the first rule first and then apply the second rule, or you can apply the second rule first and then apply the first rule.
your expression is:

applying the first rule first, your expression becomes:

applying the second rule next, your expression becomes:

applying the third rule next, your expression becomes:

we'll go back and do it again and apply the second rule first
your expression is:

applying the second rule first, your expression becomes:

applying the first rule next, your expression becomes:

simplifying this, you get:

you did not have to apply the third rule in this case.

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it looks to me like your equation would be:

anything to the 0 power is equal to 1, so the whole thing should reduce to be equal to 1.
you can go through the exercise of simplifying the expression, but if the whole thing is raised to the 0 power, then the answer has to be 1.
i show the reduced formula to be equal to:

if you follow the rules of exponentiation through, this becomes:
which becomes:
which becomes:
which becomes:
which becomes:
1.

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from what i can see, the absolute value of (x-2) is equal to the absolute value of (2-x) for all values of x.
this means your denominator will be the same and your equation of:
1/(|x-2|) + 1/(|2-x|) will become (1+1)/(|x-2|) which becomes 2/(|x-2|).
if we set this equation equal to y, we get:
y = 2/(|x-2|).
i believe that's as far as you can go with this.
a graph of that equation will look like this:

you will have an asymptote at x = 2, because when x = 2, the denominator in the equation equals 0 which makes the value of y undefined.
you can see that in the graph.
i also graphed your original equation to confirm that i substituted correctly.
that equation is:
y = 1/(|x - 2|) + 1/(|2 - x|)
that graph looks like this:

the graphs are identical, so the translation of the formula must have been correct.
if you did not recognize that abs(x-2) = abs(2-x), then what you did was in the right direction.
fyi: abs(x) is another way to show |x| and is recognized by the graphing software of algebra.com.
you would simply set that equation to y and graph it as shown below:
the equation that you derived is:
y = (abs(x-2) + abs(2-x)) / (abs(x-2)*abs(2-x))
using the algebra.com formula generator, this equation will look like this:
y =
the graph will look like this:

again, the graph is identical, so the transformation of the original equation was done correctly.
note that they indicated x < 2 which means that the graph is valid only for values of x < 2 which means all negative values of x + 0, 1, up to, but not including 2.
in interval notation, that would be shown as (-infinity, 2)
you would effectively ignore the values of x >= 2 in the graph.

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the elimination method applies multiplication factors to each equation, as required, so that when you add the equations to each other, or subtract the equations from each other, one of the variables will disappear and you will be left with one equation in one unknown which you can then solve. once you have solved for one of the variables, you can then use that value to help solve for the other variable.
your equations are:
3x + 4y = 5
2x + y = 1
if you multiply the second equation by 4, you will be able to eliminate the y from the equation and you can then solve for x.
when you multiply an equation, you have to multiply both sides of the equation in order to preserve the equality.
multiplying the second equation by 4 gets you:
3x + 4y = 5 (first equation)
8x + 4y = 4 (second equation multiplied by 4)
if you subtract the first equation from the second equation, you will be left with:
5x = -1
divide both sides of this equation by 5 and you get:
x = -1/5
you can now substitute for x in either of the 2 original equations in order to solve for y.
using the first equation of:
3x + 4y = 5
substitute (-1/5) for x to get:
3*(-1/5) + 4y = 5
simplify to get:
(-3/5) + 4y = 5
add (-3/5) to both sides of the equation to get:
4y = 5 + (3/5) which becomes:
4y = 28/5
divide both sides of this equation by 4 to get:
y = 7/5
the 2 values for x and y are:
x = -1/5
y = 7/5
substitute for x and y in the first original equation of:
3x + 4y = 5 to get:
3*(-1/5) + 4*(7/5) = 5 which becomes:
-3/5 + 28/5 = 5 which becomes:
25/5 = 5 which becomes:
5 = 5, confirming the values for x and y are solutions for the first equation.
substitute for x and y in the second original equation of:
2x + y = 1 to get:
2*(-1/5) + (7/5) = 1 which becomes:
-2/5 + 7/5 = 1 which becomes:
5/5 = 1 which becomes:
1 = 1, confirming the values for x and y are solutions for the second equation.
those are your answers:
x = -1/5
y = 7/5

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x = number of liters of 1% solution.
y = number of liters of 9% solution.
you want a total of 5 liters of 3.24% solution.
your first equation is:
x + y = 5
this tells you that the total amount of solution will be 5 liters.
your second equation is:
.01*x + .09*y = .0324*5
this tells you that the mix of x liters of 1% solution + y liters of 9% solution will equal 5 liters of 3.24% solution.
you need to solve these 2 equations simultaneously to get your answer.
the equations are:
x + y = 5
.01*x + .09*y = .0324*5
simplify the second equation to get:
.01*x + .09*y = .162
your 2 equations are now:
x + y = 5
.01*x + .09*y = .162
solve for x or y in the first equation and then substitute in the second equation.
from the first equation, solving for y gets:
y = 5 - x
substituting for y in the second equation gets:
.01*x + .09*(5-x) = .162
since you have reduced the second equation from 2 unknowns to one unknown by applying the substitution from the first equation, you can solve for x in the second equation.
simplify the second equation to get:
.01*x + .09*5 - .09*x = .162
combine like terms and simplify further to get:
-.08*x + .45 = .162
subtract .45 from both sides of this equation to get:
-.08*x = .162 - .45
simplify this to get:
-.08*x = -.288
divide both sides of this equation by -.08 to get:
x = -.288 / -.08
simplify this to get:
x = 3.6
use your first equation to derive that y will be equal to 1.4
you have:
x = 3.6 liters
y = 1.4 liters
x + y = 5 becomes 3.6 + 1.4 = 5 which becomes 5 = 5, confirming the values for x and y solve the first equation.
substituting in your second equation of:
.01*x + .09*y = .162 gets you:
.01*3.6 + .09*1.4 = .06 which becomes .036 + .126 = .162 which becomes .162 = .162, confirming the values for x and y solve the second equation.
the values of 3.6 for x and 1.4 for y have solved both equations simultaneously, so they're good.
your answer is:
you need 3.6 liters of the 1% solution and 1.4 liters of the 9% solution to make 5 liters of 3.24% solution.

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you are traveling at 200 miles per hour.
divide that by 60 and you will be traveling 200/60 = 3 and 1/3 miles per minute.
multiply that by 60 and you get back to 200 miles per hour.
there are 60 minutes in an hour.

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here's a good reference that will help you understand convex and concave polygons.
http://www.mathopenref.com/polygonconvex.html
the convex polygon will not have an internal angle greater than 180 degrees.
the concave polygon will.
the figure shown in your jpg file looks very much like a concave polygon.
there is no such a thing as a regular concave polygon (see the reference).
your figure looks very much like a concave polygon.
from wikipedia:
A simple polygon that is not convex is called concave[2] or reentrant.[3] A concave polygon will always have an interior angle with a measure that is greater than 180 degrees.
It is always possible to cut a concave polygon into a set of convex polygons. A polynomial-time algorithm for finding a decomposition into as few convex polygons as possible is described by Chazelle & Dobkin (1985).[4]
end of from wikipedia:

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to graph this, we make y = 8500*x where x is the interest rate.
since the graph shows x as an integers from 1 to 20, then we need to modify the equation to show as y = 8500 * .01*x.
this gets us the decimal equivalent of the percent.
10% = .1
1% = .01
etc.
the graph will look like this:

i drew a horizontal line at y = 1020 to show you that when x = .12, y = 1020.
8500 * .12 = 1020 which is the simple interest on 8500 when the rate is equal to .12, or 12%.
if you trace a vertical line at x = 12, you will see that it intersect the line of the equation at y = 1020.

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your original equation is shown as:
12-{3+[4-5(6-2+7)+1]}
after converting all symbols to ( and ) and indicating multiplication with *, your equation appears to be:
12 - (3 + (4 - 5 * (6 - 2 + 7) + 1))
the order of arithmetic operations is as follows:
resolve inner parentheses first, then next inner, then next inner, until you have resolved the expression.
the expression itself is assumed to be within an outer set of parentheses (not shown).
within each set of parentheses work from left to right.
if 2 sets of parentheses are on the same level, resolve the left one first and then the right one.
working from left to right within each set of parentheses:
exponents and roots are done first.
multiplies and divides are done next.
adds and subtracts are done next.
your expression is:
12 - (3 + (4 - 5 * (6 - 2 + 7) + 1))
expression within inner set of parentheses appears to be (6 - 2 + 7) so that is resolved first to get 11.
your expression becomes 12 - (3 + (4 - 5 * 11 + 1))
expression within inner set of parentheses now appears to be (4 - 5 * 11 + 1)
working from left to right within that set of parentheses, the multiplication is done first to get (4 - 55 + 1).
the adds and subtracts within that set of parentheses are done next, working from left to right, to get 4 - 55 = -51 + 1 = -50.
your expression becomes 12 - (3 + (-50)) which becomes 12 - (3 - 50).
your next set of inner parentheses now appears to be (3 - 50).
resolve this to get (-47)
your expression becomes 12 - (-47) which becomes 12 + 47 which becomes 59.
your answer should be 59.
i'll go through the process again without all the verbiage so you can see how the order of operations progresses starting with:
12 - (3 + (4 - 5 * (6 - 2 + 7) + 1))
12 - (3 + (4 - 5 * 11 + 1))
12 - (3 + (4 - 55 + 1))
12 - (3 + (-51 + 1))
12 - (3 + (-50))
12 - (3 - 50)
12 - (-47)
12 + 47
59

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3x - 7 = 0
3x = 7
x = 7/3
that just solves for x but doesn't answer the question.
your solutions are:
A.
adding 0 to 3x - 7 results in 3x - 7 so that isn't a solution.
B.
adding 3x + 7 to 3x + 7 results in 6x + 14 so that isn't a solution.
C.
adding -3x + 7 to 3x + 7 results in 14 so that isn't a solution.
D.
adding -3x - 7 to 3x + 7 results in 0 (bingo !!!!!)
your equation becomes (3x+7) + (-3x-7) = 3x + 7 - 3x - 7 which results in 0 after you combine all like terms.

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let x = original price of the door.
the sale price of the door is equal to x -.2x = .8x
the total cost is equal to 43.99 which is the sum of the sale price of the door plus the shipping and handling.
this leads to:
43.99 = .8x + 4.95
subtract 4.95 from both sides of this equation to get:
43.99 - 4.95 = .8x
simplify this to get:
39.04 = .8x
divide both sides of this equation by .8 to get:
39.04 / .8 = x which leads to:
x = 39.04 / .8 = 48.8
the original price of the door is $48.80
20% off the original price leads to a sale price of 39.04
add shipping and handling of 4.95 to get a total price of 39.04 + 4.95 = $43.99

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the room is 12 feet 6 inches by 10 feet 4 inches.
converting everything to feet, you get:
the room is 12.5 feet by 10.33333333 feet.
when you multiply 12.5 by 10.3333333, you get 19.16666666666 square feet.
that fractional part of .16666666666666 looks like the fraction 1/6.
i'll assume that's true for now and confirm it later.
assuming it's true, then the answer is 129 + 1/6 square feet.
converting that to an improper fraction means putting everything under a common denominator.
since 129 is equivalent to 6*129/6, then you get the answer being 774/6 + 1/6 square feet which comes out to be 775/6 square feet.
you could have also solved using improper fractions to start with.
we'll start again and solve it that way.
the room is 12 feet 6 inches by 10 feet 4 inches.
that's equivalent to 12 and 1/2 feet by 10 and 1/3 feet.
12 and 1/2 feet is equivalent to (12*2)/2 + 1/2 feet which is equivalent to 24/2 + 1/2 feet which is equivalent to 25/2 feet.
10 and 1/3 feet is equivalent to (30/3) + 1/3 feet which is equivalent to 31/3 feet.
the dimensions of your room are now:
25/2 feet by 31/3 feet.
the area is length times width which becomes:
25/2 * 31/3 which results in (25*31)/(2*3) square feet.
this is equivalent to 775/6 square feet.
i don't believe it can be reduced further than that since 775 is not divisible by 2 or 3 or 6, so that has to be your answer (if i am correct).
since i got the same answer 2 different ways, i'm reasonably confident it is.
to confirm that .1666666666..... is equivalent to 1/6, i simply use my calculator to divide 1 by 6 and get the decimal answer of 1.6666666667.
the difference is in rounding.
1.66666666667 is really 1.66666666666666666666666666666......... endlessly, so the answer is rounded to the last decimal place that the display on the calculator can show.

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a rational number is a number that can be expressed as the ratio of 2 integers.
examples:
5 is a rational number because it can be expressed as 5/1 or 10/2 or 15/3, etc.
.25 is a rational number because it can be expressed as 1/4, or 2/8, or 3/12, etc.
your problem states:
Find two rational numbers such that the second number is five more than three times the first number. The sum of numbers is 19.
let x and y be your rational numbers.
the second number is 5 more than 3 times the first number leads to the equation:
y = 3*x + 5
the sum of the numbers being 19 leads to the equation:
x + y = 19
you need to solve these equations simultaneously to get your answer.
since y = 3*x + 5 in the first equation, use this value of y to substitute for y in the second equation to get:
x + y = 19 becomes x + (3*x + 5) = 19
simplify by removing parentheses to get:
x + 3x + 5 = 19
combine like terms to get:
4x + 5 = 19
subtract 5 from both sides of this equation to get:
4x = 14
divide both sides of this equation by 4 to get:
x = 14/4
since you know the value of x, you can now solve for the value of y.
use the equation x + y = 19 to get:
14/4 + y = 19
subtract 14/4 from both sides of this equation to get:
y = 19 - 14/4
put everything on the right side of this equation under a common denominator of 4 to get:
y = (4*19)/4 - 14/4 which becomes y = (76 - 14)/4 which becomes y = 62/4
your answer appear to be:
x = 14/4 which can be reduced to 7/2.
y = 62/4 which can be reduced to 31/2.
they are both rational numbers so all we need to do is confirm that they are the correct solutions for both equations.
the first equation is:
y = 3*x + 5
substituting for x and y, we get:
31/2 = 3 * (7/2) + 5
this becomes 31/2 = 21/2 + 10/2 which becomes 31/2 = 31/2, which is true, confirming the values of x and y are solutions for the first equation.
the second equation is:
x + y = 19
substituting for x and y, we get:
31/2 + 7/2 = 19 which becomes 38/2 = 19 which becomes 19 = 19, which is true, confirming that the values of x and y are solutions for the second equation as well.
since the values of x and y are solutions for both equations simultaneously, then they are good.
the first number is 7/2
the second number is 31/2
the sum of the first and second number is equal to 19 (7/2 + 31/2 = 38/2 = 19).
the second number is equal to 3 times the first number plus 5 (31/2 = 21/2 + 5 = 21/2 + 10/2 = 31/2).

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let x = jam's age now.
let y = dindo's age now.
in 2 years jam will be as old as dindo is now.
this means that x + 2 = y
in 26 years dindo will be twice as old as jam is now.
this means that y + 26 = 2 * x
you have 2 equations that need to be solved simultaneously.
the first equation is:
y = x + 2
the second equation is:
y + 26 = 2 * x
the trick in solving 2 equations in 2 unknowns is to reduce the number of unknowns and equations to 1 which can then be solved. after you solve for the 1 unknown, then you can solve for the other.
in this problem, we'll use substitution as the method to reduce the 2 equations in 2 unknowns to 1 equation in 1 unknown.
the first equation says that y = x + 2.
we'll substitute for y in the second equation to get:
y + 26 = 2 * x becomes:
(x + 2) + 26 = 2 * x
that last equation is now 1 equation in 1 unknown.
simplify it to get:
x + 28 = 2 * x
subtract x from both sides of the equation to get:
28 = x
this is the same as:
x = 28
you now know what x is and you can solve for y.
use the first equation to get:
y = x + 2 which becomes:
y = 30
you now should have both x and y and you can confirm them as being good or not good by substituting in the original equations.
the first equation is y = x + 2
since y = 30 and x = 2, this equation is true and the values of x and y are confirmed as good.
the second equation is y + 26 = 2 * x
y + 26 = 30 + 26 = 56 and 2 * x = 2 * 28 = 56, so the equation is true and the value of x and y are confirmed as being good for the second equation.
since the value of x and y are confirmed as good in both equations, then the value of x and y have simultaneously solved both equations.
the answer is:
x = 28
y = 30
this means that jam is 28 and dindo is 30.
in 2 years jam will be 30 which is how old dindo is now.
in 26 years dindo will be 30 + 26 = 56 years old which is 2 * how old jam is now (jam is now 28 years old).

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The problem is:
The measure of an angle and its complement differ by 22. Find the angle and its complement.
the answer is:
if the angle is equal to x, then the complement of the angle is equal to 90 - x.
that's because the sume of an angle plus its complement must always be equal to 90 degrees.
the problem states that the angle and its complement differ by 22.
this means that, if the angle is equal to x, then the angle's complement is either equal to x + 22, or x - 22.
let's assume that the complement of the angle is equal to x + 22.
since the angle plus its complement must always be equal to 90 degrees, then we get the equation:
x + (x + 22) = 90
this results in 2x + 22 = 90.
subtract 22 from both sides of this equation to get:
2x = 68
divide both sides of this equation and you get x = 34.
this makes x + 22 = 56
the sum of 34 and 56 = 90 so we're good.
let's assume that the complement of the angle is equal to x - 22.
since the angle plus its complement must always be equal to 90 degrees, then we get the equation:
x + (x - 22) = 90
this results in 2x - 22 = 90.
add 22 to both sides of this equation and you get 2x = 112
divide both sides of this equation by 2 and you get x = 56
this makes x - 22 = 34.
the sum of 56 + 34 = 90 so we're good.
either way you get an angle of 34 degrees and an angle of 56 degrees.
if you pick your angle as being 34, then the complement is 56.
if you pick your angle as being 56, then the complement is 34.
the angle plus its complement must always be equal to 90 degrees.
that's the rule that allows you to solve this problem.

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The equation is
the expression within the square root sign can't be negative.
since we are dealing with x^2 within the square root sign, then any real number can be substituted for x and the result will be positive.
therefore, the domain of this equation is the set of all real numbers.
to give you some examples:
x = -5, then y = sqrt(27) which is a positive number within the sqrt sign.
x = 0, then y = sqrt(2) which is a positive number within the sqrt sign.
x = 5, then y = sqrt(27) which is a positive number within the sqrt sign.
the domain is the set of all real numbers.
SINCE YOU WERE ONLY ASKED TO FIND THE DOMAIN, YOU CAN STOP HERE.
IF YOU ARE INTERESTED, YOU MAY CONTINUE TO THE DISCUSSION ON THE RANGE SHOWN BELOW:
the range (value of y, given the value of x) is the set of all real numberS >= sqrt(2).
that's because the number within the square root sign will never be less than 2.
a graph of this equation is shown below:

note that the result of this equation is the principal, or positive square root of the expression only.
that's because the radical sign is part of the equation.
even though the negative square root is also valid, it is not part of the range of this equation because of the existence of the radical sign, which forces the answer to be the positive square root only.
a reference that explains this sometimes confusing situation is shown below.
http://www.helpalgebra.com/onlinebook/roots.htm
because the radical sign implies the positive, or principal, square root only, then, when you are asked to solve the equation y = , the answer is plus 2, and when you are asked to solve the equation y^2 = 4, then the answer is plus or minus 2.
in the first case, the radical sign forced the answer to be the principal, or positive square root only. in the second case, the fact that there was no radical sign in the equation allowed the answer to be both the principal and the negative square root.
if your equation were:
y^2 = x^2 + 2, then the answer would have been y = plus or minus and the graph would have looked like this:

in that case, the domain (the value of x) would have still been the set of all real numbers, but the range would have been x >= sqrt(2) or x <= -sqrt(2).
note the special case of y = in the reference.
the reference states that y = = 5
= |-5| = 5
without this rule, then the square root of (-5)^2 could have been interpreted as being equal to (-5) because the square root of x^2 = x. the rule forces the answer to be the principal root only, even through the original root was negative.

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you have the right idea.
your equation is:
6y + 11x = -11
subtract 11x from both sides of the equation to get:
6y = -11x - 11
divide both sides of the equation by 6 to get:
y = (-11x/6) - (11/6)
this can also be written as:
y = (-11/6)*x - (11/6)
your equation is now in the slope intercept form of the equation of a straight line.
the slope is equal to (-11/6) and the y intercept is equal to -(11/6)
you now need to find 3 values of x that will result in 3 values of y and then you need to graph the 3 points.
they will form a straight line.
pick x = 0 as your first number.
you will get:
when x = 0, y = -(11/6)
pick x = 1 as your second number.
you will get:
when x = 1, y = (-11/6)*1 - (11/6) which becomes y = (-11/6) - (11/6) which becomes y = (-11-11)/6 which becomes (-22/6).
pick x = 2 as your third number.
when x = 2, y = (-11/6)*2 -(11/6) which becomes y = (-22/6) - (11/6) which becomes y = (-22-11)/6 which becomes y = (-33/6).
you have 3 points.
you graph these 3 point pairs.
they are:
(x,y) equals:
(0,-11/6)
(1,-22/6)
(2,-33/6)
a graph of your equation looks like this:

notice that when x = -1, y = 0
that give you 4 points to explore.
they are:
(x,y) equals:
(-1,0)
(0,-11/6)
(1,-22/6)
(2,-33/6)
notice that when x moves up 1, y moves down 11/6.
that's because the slope of the equation is (-11/6)
the slope is the change in the value of y divided by the change in the value of x.
y goes down 11/6 units every time x goes up 1 unit.
that's a slope of (-11/6) / 1 which equals (-11/6).
if i didn't mention it before, the general form of the slope intercept form of the equation of a straight line is:
y = mx + b
m equals the slope
b equals the y intercept (value of y when x = 0)
your equation is:
y = -(11/6)x - (11/6)
m = slope = -(11/6)
b = y intercept = value of y when x = 0 = -(11/6)

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this is a pain in the butt because you don't get a clean answer, but i confirmed that the answer is correct, so i must be doing something right.
here's how you would solve this problem.
you start with:
log(1 + x) + log(2 + x) = 2
the concepts you will use to solve this are:
concept number 1:
log(a) + log(b) = log(a * b)
concept number 2:
y = log(x) if and only if x = 10^y
note that log(x) could also be written as log(10,x) which means log of x to the base 10.
if you are dealing with the base of 10, then you don't need to show the 10 which is why log(10,x) can be shown as log(x).
if it's any other base, you would need to show the base.
for example:
log of 20 to the base of 2 would be shown as log(2,20).
the general form of this statement would be log(b,x) means log of x to the base b.
under concept number 2, please be advised of the following:
y = log(x) if and only if x = 10^y would also be written as:
y = log(10,x) if and only if x = 10^y.
the general form of this concept would be:
y = log(b,x) if and only if x = b^y
the b represents any base.
going back to your problem:
you started with the equation:
log(1 + x) + log(2 + x) = 2
using concept number 1, you transform this equation to:
log( (1 + x) * (2 + x) ) = 2
using concept number 2, you transform this equation to:
2 = log( (1 + x) * (2 + x) ) if and only if 10^2 = (1+ x) * (2 + x)
solve this equation and you have your answer.
this is a basic quadratic equation.
multiply out the factors on the right side of this equation and you get:
10^2 = x^2 + 3x + 2
this is equivalent to:
x^2 + 3x + 2 = 100
subtract 100 from both sides of this equation to get:
x^2 + 3x - 98 = 0
this can't be factored by eye, so you need to resort to the quadratic formula in order to solve this quadratic equation.
the quadratic formula is:
x = ( (-b) +/- sqrt(b^2 - 4ac) ) / (2a)
your quadratic equation is:
x^2 + 3x - 98 = 0
this is in standard form of ax^2 + bx + c = 0
this means that:
a = 1
b = 3
c = -98
substituting in the quadratic formula gets us the following:
x = 8.512492197
or:
x = -11.5124922
those are your answers.
if you substitute either of those values for x in your original equation, you will see that the equations are true, confirming these answers are good.
for example:
using the 8.5 number, your original equation of:
log(1 + x) + log(2 + x) = 2 becomes:
log(1 + 8.512492197) + log(2 + 8.512492197) = 2 which becomes:
log(9.512492197) + log(10.512492197) = 2 which becomes:
.978294314 + 1.021705686 = 2, confirming that the equation is true when the value of x is replaced with 8.512492197.
note that i used my calculator to get the log of 9.512492197 and to get the log of 10.512492197.
my statement about this being a pain in the butt is because the factorization of the quadratic equation wasn't clean and we had to deal with fractional numbers requiring the use of the calculator to find the answer. I stored intermediate results in memory so I didn't have to type all the decimal places that i showed you here.
so, that's how it's done.
the basic concepts i showed you are what you need to use to solve this, plus you need to know the quadratic formula to solve for the quadratic equation.
plus you needed to recognize that you had a quadratic equation in the first place.

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your equation is:
f(x) = (x - 2)(x + 6)
multiply those factors together and you get:
x^2 + 4x - 12
that looks a lot like a quadratic equation.
a graph of that equation is shown below:

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here's how it works:
your equation is:
9 log w + 4 log z - log (x - 5)
you have 3 basic concepts working here:
concept number 1:
log (a^b) = b*log(a)
concept number 2:
log (a * b) = log(a) + log(b)
concept number 3:
log (a / b) = log(a) - log(b)
using these concept your equation will be transformed as follows:
9 log w + 4 log z - log (x - 5)
using concept 1, you get:
log(w^9) + log(z^4) - log(x - 5)
working from left to right:
using concept 2, you get:
log(w^9 * z^4) - log(x - 5)
using concept 3, you get:
log( (w^9 * z^4) / (x - 5) )
This looks a lot like it matches selection C) log w^9*z^4/x-5

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a, b, c, and d are all positive numbers.
assume a = b
assume c < d
add a to both sides of the equation of c < d to get:
a + c < a + d
this is a valid statement based on the properties of arithmetic that state that you can add the same amount to both sides of an inequality and you will preserve the inequality.
now, since you are given that a = b, you can replace one of the a's in your equation with b and the equation will remain the same.
your equation of:
a + c < a + d becomes:
a + c < b + d
conversely:
a is equal to b
in the equation we just derived of a + c < b + d, you can replace b with a to get:
a + c < a + d
you can subtract a from both sides of this equation to get:
c < d
this proves that if a = b and c < d, then a + c < b + d
you are using some basic properties of arithmetic that prove this.
those properties are given and don't have to be proved.
they can be assumed to be true.
here's a reference
http://www.allaboutcircuits.com/vol_5/chpt_4/2.html
here's another reference
http://hotmath.com/hotmath_help/topics/properties-of-equality.html
here's another reference
http://hotmath.com/hotmath_help/topics/properties-of-inequality.html
the property that is used in the proof is the addition property.
with this last reference, the < symbol with a / superimposed on it means NOT smaller than, while the > symbol with a / superimposed on it means NOT greater than.
i can only show it as /> and /<.
the statement x /> x means that x is not greater than x, which is intuitively obvious because x = x.
read the references.
read the proof.
hopefully it will all make more sense to you after you're done.
an example may help clarify the concept.
let c = 4 and let d = 5
clearly c < d
let a and b both equal to 9.
start with c < d
substitute 4 for c and 5 for d to get:
4 < 5
add a to the left sides of this equation and add b to the right side of this equation.
you get:
a + 4 < b + 5
substitute 9 for a and 9 for b to get:
9 + 4 < 9 + 5
combine like terms to get:
13 < 14
you can also prove this in a reverse fashion by assuming that the statement is not true.
this would be:
if a = b and c < d, then a + c is not smaller than b + d
that means a + c is either equal to b + d or is greater than b + d.
let's make our statement to read:
if a = b and c < d, then a + c >= b + d
if we can disprove this statement, then the only other option is that it is smaller than b + d.
we start with:
a = b (given)
c < d (given)
our equation is:
a + c >= b + d
since b = a, we can replace b with a to get:
a + c >= a + d
if we subtract a from both sides of this equation, we get:
c >= d
but, we are given that c < d, so this equation must be false and we are left with the only other option left, which is that a + c < b + d.
once again, we can use an example:
let c = 4 and d = 5 and let a = 9 and b = 9.
we start with a + c >= b + d and we substitute to get:
9 + 4 >= 9 + 5 which becomes:
13 >= 14 which is false.
again, the contradiction based on the false assumption leading to the conclusion that:
if a = b and c < d, then a + c < b + d

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a rectangle that is 240 by 30 by 5 cubic feet is equal to 240 * 30 * 5 = 36000 cubic feet.
from an internet search, i have determined that 1 cubic foot is equivalent to 7.48051948 gallons.
based on that information, 36000 cubic feet is equivalent to 36,000 * 7.48051948 = 269,298.7013 gallons.

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rate * time = distance
time = 5 hours
distance = 300 miles
rate * 5 = 300
rate = 300 / 5 = 60 miles per hour.
5 hours times 60 miles per hour equals 5 * 60 = 300 miles
natasha would have to average 60 miles per hour in order to be able to drive 300 miles in 5 hours.