what multiplies to get -270 and adds to get -3?

1)xy=-270
2)x+y=-3


y=-x-3
x(-x-3)=-270
x^2+3x-270=0
(x-15)(x+18)=0

so, 15 and -18

if 1/2 of an integer is added to 1/4 to the next consecutive integer the sum is 22.find the integer.

multiplying by 4...

write the missing digit to make 22_2 divisible by 9.

It is well known that if the digits of a number add to a multiple of 9 then so does the number. We look for a 1-digit number such that

2+2+n+2 adds to a multiple of 9
Since our number is 1-digit we need to use 3
2232/9=248
:)

the smallest number greater than 200 that is divisible by both 3 and 9

It is well known that if the digits of a number add to a multiple of 3 then so does the number (the same property holds for 9). We look for a number such that

2_ _ is a minimum, and divides 9

so we can see the least number possible must be 207

:)

How can I show a real number on a graph?

The graph for the real numbers is the number line.

1) decide on a scale - say for example 1 cm = 1 unit
2) draw and label a number line
3) use the magnitude of the number to determine how far from 0 to place a point
4) use the sign of the number to determine whether to plot it left (for negative) or right of 0

If the product of the integers a, b, and c is 1, then what is the smallest possible value of (a+1)(b+1)(c+1)?

Expanding and simplifying...
(a+1)(b+1)(c+1)=
(a+1)(bc+b+c+1)=
abc+ab+ac+bc+a+b+c+1=
ab+bc+ac+a+b+c+2
Since their product is 1, the only possibilities are a,b,c={-1,1} (why?)
Furthermore, there must be an odd number of positives. To find the minimum we can ignore the case a=b=c=1 - that would be the maximum.
Case1
if say a=b=1, c=-1, then
ab+bc+ac+a+b+c+2=
1-1-1+1+1-1+2=
2
Case2
if say c=1, a=b=-1, then
ab+bc+ac+a+b+c+2=
1-1-1-1-1+1+2=
0
Minimum value is 0

What do you do to check whether a number is rational or irrational? Give an example of a rational and irrational number.

If a number can be written as a fraction its decimal expansion will eventually begin to repeat and it is rational. If not it is irrational. Examples: pi is irrational and 0.125=0.12500000...=1/8 is rational.

Graph the number below on a number line. sqrt(11)

3^2 < 11 < 4^2
so that means
3 < sqrt(11) < 4
and since 11 is closer to 9, sqrt(11) is closer to 3.
This process can be refined as much as we like to get as close to sqrt(11) as our resources allow, but since sqrt(11) is irrational, we can never locate it's exact position

find the largest four digit number which is divisible by 5,8,and 9

our number must be a multiple of...
(5)(8)(9)=360
The greatest 4-digit number is 9999, so...
9999/360=27 31/40
so take 27...
360*27=9720

find the smallest square number that is divisible by 6, 18 and 21?

6=2*3
18=2*3^2
21=3*7 

To be a square we need all prime factors squared (at least once in general):
n=((2)(3)(7))^2=1764

The number 18 can be written as the sum of 9 consecutive integers.What's the product of these integers?

which when solved leads to x=-2
so our product is

What is 0.154545454545454(repeating decimal) as a fraction? And How do you rewrite repeating decimals as a fraction?

n=0.154545454545454...
10n=1.5454545454545...
1000n=154.545454545...
1000n-10n=153
990n=153
9*110n=9*17
n=17/110

there is a three digit number one digit is prime another is a square number the last one is not prime or square. the number is not divisible by three. what is the greatest possible value of the number

we try the first digit as 9 because it is a perfect square and the greatest 3-digit numbers have 9 in their thousands place
the last one is not prime, so the middle number must be the prime so we choose 7 (greatest prime less than 9) 
6 is not a prime, it is our only last digit possible if we require our number to not be a multiple of 3 
so our number is 976 

What multiplication problem can give you negative 18, but when subtracted, gives you a negative 1?

a*b=-18 
a-b=-1 

which we rewrite as 
a=b-1 

then we substitute 
(b-1)b=-18 
Now we solve for b 

b^2-b+18=0 

which has the following solution 
 
So as you can see there is no real pair of numbers that can do both at the same time but if we allow the numbers to be complex, then it is possible 

:)

An integer plus its reciprocal equals to 29/10. Solve.

which has solutions
{2/5 , 5/2}

How do you write 1.86 as a repeating decimal, with the repeat sign over the six, turned into a simple fraction?

100n-10n=(186.6666...)-(18.6666...)

How can I rewrite the expression x+(y+-z)using the associative property of addition?

Associative property tells us that we can add in any order and the parenthese tells us where we are choosing to add first: 

x+(y+-z)=(x+y)+(-z)

how do you find five consecutive odd integers such that the sum of the first and the fifth is one less than three times the fourth

consecutive integer will be these...
1:n
2:n+2
3:n+4
4:n+6
5:n+8 
n + (n+8) = 3(n+6)-1
2n+8=3n+18-1 
-n=17-8
n=-9 
they are... 
-9, -7, -5, -3, -1 

The sum of the digits of a two-digit number is 9. If the digits are reversed, the new number is 27 less than the original. Find the original number.

We don't need algebra... 
81 is too great
72 as well...so...
63-36=27

Find two consecutive positive intergers the sum of whose squares is 85.

Careful thought will convince you that this task is impossible. The square of an odd number is also odd. This implies that the sum of the squares of any even number of odd numbers must be even. But 85 is odd...this is a contradiction to the the above true statements :)