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Think of it this way. Imagine that you are walking to town. You have two choices. You can take one step toward the town. This is a + move. Or you can get turned around and go away from town. This is a - move. Now you look at your map and if you see you are going toward town you continue with the + move. If you discover you are going in the wrong direction you make a change this is a - move.
If the two numbers have different signs, then the result will be negative. If the two numbers have the same signs, the result will be positive. This assumes all the numbers are non-zero.
Then you are adding two rational numbers with different signs! No big deal!
You get a positive answer when multiplying two alike signs because since its an even number of positive signs, the number will be positive. For example, -2 x -4= 8 since there are an even amount of negative signs.
The value of the quotient of two integers with different signs is the same as if the signs were the same. Because the numbers have different signs, the quotient is negative.