The rules for integer operation

 ADDITION
*When adding two integers with the SAME SIGN (both positive, both negative), ADD the integers and KEEP the SIGN.

Examples:
4 + 8 = 12

-6 + (-10) = -16

*When adding two integers with DIFFERENT SIGNS (one negative, one positive):
    1. Take the absolute value of both integers.
    2. Take the SIGN of the integer with the GREATER absolute value.
    3. SUBTRACT the integers.
    4. Your answer will be the DIFFERENCE of the integers and the sign resulting from step #2.

Examples:
10 + (-12)
      1. Take the absolute value of the integers:
         |10| = 10 (positive 10 is 10 units away from zero)
         |-12| = 12 (negative 12 is 12 units away from zero)
      2. Take the sign of the integer with the greater absolute value.
          |10| < |-12|
           10  < 12
          THE ANSWER WILL BE NEGATIVE.
       3. Subtract the integers.
          12 - 10 = 2
       4. The answer is: -2
          10 + (-12) = -2

-7 + 14
 |-7| < or > |14|
  7 < 14
 The answer will be positive (14 is positive in the original problem).
  14 - 7 = 7
  Therefore, -7 + 14 = 7


SUBTRACTION
*When there is a "double negative" then you must change the signs to positive.  Like in English, in math a double negative "cancels."  In English, to say "I am not unhealthy" means "I am healthy."  The first has a double negative, but we can say the same thing as a "positive."
NOTE: NEVER CHANGE THE SIGN OF THE FIRST INTEGER.

Examples:
17 - (-8)
17 + (+8)
17 + 8 = 25

-3 - (-6)
-3 + (+6)
-3 + 6
(Using the rules for addition, above, we can solve the rest of the problem by doing the following)
|-3| < or > |6|
3 < 6, therefore the answer will be positive
6 - 3 = 3
Therefore: -3 - (-6) = 3

*When you are subtracting two positives, and, say, the second integer has a higher value, you can re-write the problem as an addition problem.

Example:
4 - 15

We can re-write this as: 4 + (-15) and solve using the rules above (see addtion).

This works because subtraction is just like adding a negative number (or adding the "opposite," also called the "additive inverse," like 15 is the additive inverse of -15 and -15 is the additive inverse of 15).


Multiplying and dividing integers is easier and you can use the same rules for both.  I'm going to go through them individually, with examples, and then generalize the rules.

MULTIPLICATION
*The product of two POSITIVE integers is POSITIVE.
Example:  10 x 5 = 50

*The product of two NEGATIVE integers is POSITIVE.
Example: -4 x -12 = 48

*The product of a POSITIVE integer AND a NEGATIVE integer is NEGATIVE.
Example: 2 x -14 = -28

DIVISION:
*The quotient of two POSITIVE integers is POSITIVE.
Example: 36/9 = 4

*The quotient of two NEGATIVE integers is POSITIVE.
Example: -15/(-3) = 5

*The quotient of a POSITIVE integer AND a NEGATIVE integer is NEGATIVE.
Example: -12/4 = -3

Generally (for multiplication, division):
*If the signs are the SAME, your answer is POSITIVE.
*If the signs are DIFFERENT, your answer is NEGATIVE.