You did something wrong.
Algebra is nothing more than a way of doing arithmetic in general. When you multiply both sides of an equation by (x + 10), you’re doing it for all possible values of x at once. This works because multiplication is a very predictable thing. However, you still need to think about what you’re doing at every stage.
Let’s break down what you’ve done.
(x+10)/2
---------- = 1/2
x+10
This is fine. One number, divided by another number, is equal to 1/2. The first thing to notice is that if x = -10, this is not a well defined expression, because it is equivalent to 0/0, and 0/0 doesn’t have a value.
Now you multiply both sides of the equation by (x+10). Very well, we get the following:
(x+10)/2
---------- * (x+10) = (x+10)/2
x+10
which can be rewritten as
x+10
(x+10)/2 * ------ = (x+10)/2
x+10
At this point, you think to yourself “any number divided by itself is 1” so I can replace (x+10)/(x+10) by 1, which it is equal to. Of course, it isn’t true that any number divided by itself is 1, because 0/0 is not any number at all–it’s not defined. So, as long as x+10 is different from 0, you can move on to your next step. However, if x+10 is actually 0, you cannot.
The key here is that algebra is a way to do arithmetic with unknown quantities. If you are careful, you aren’t “moving things from one side to the other” and you don’t ever “cancel the sevens.” That’s just shorthand for doing several steps in a row. Trying to “cancel” terms leads people to no end of trouble, because they have forgotten (or never learned) that what they are doing is replacing one fraction (x/x) by another fraction (1) which it is equal to.
In the example above, the fraction (x+10)/(x+10) is the same as 1, so long as x+10=/=0. And that’s where your mistake lies.