I don't mean to disagree with the other explanations offered. Because, there may still be confusion, I offer this....
The basic concepts in solving an equation in one variable is to do the same thing to both sides of the equation, selecting what to do, so that you finally end up with "one" variable on one side and a single number on the other side.
4x - 9 = 3 - 4x
We are trying to get 1x on one side of the equation, so, we must get rid of one of the "4x's". If you choose to eliminate the -4x from the right side of the equation, you will add +4x to both sides. If you choose to eliminate the +4x from the left side, you will subtract 4x (or add -4x) from both sides. Let's do the former.
4x - 9 + 4x = 3 - 4x + 4x.
Now, simplify that.
8x - 9 = 3
Now, we need to get rid of the -9 from the left side. We do so by adding 9 to both sides.
8x - 9 + 9 = 3 + 9
Now, simplify.
8x = 12
We now know the value of 8 of those x's, but we want to know the value of only one of them. Therefore, if we have eight of them, to get one of them we will divide by 8. If we divide the left side by 8 we must divide the right side by 8. (We have made the left side one-eighth as large as it was, so, to maintain the equality, we must make the right side one-eighth as large as it was.)
\(\displaystyle \frac{8x}{8}=\frac{12}{8}\)
Simplify and reduce if possible.
\(\displaystyle x=\frac{12}{8}=\frac{3}{2}\)
I leave the check to you as explained above.