How many solutions exist for the given equation?
1/2
(x + 12) = 4x – 1

There are two ways to look at this, one is by solving this and the other is by looking at the variable

First, if we want to solve it, we would take our equation, (i like decimals) .5(x+12)=4x-1 and distribute the .5 to both things in the parentheses. when we do that we get .5x+6=4x-1. now we get our variables to the same side by subtracting .5x from both sides. When we do this, we are left with 6=3.5x-1. next we add 1 to each side and get 7=3.5x. Now we divide each side by 3.5, and get x=2. Now we have to plug it into the original equation and get .5(2+12)=4(2)-1.   when we simplify this we get .5(14)=7, which goes to 7=7. this is a true statement, so we know that x=2.  since we only have one value for x, we know the answer is that only one solution exists. 

Another way that is simpler, but not always accurate, is to look at the variable. There can only be as many solutions for x as the highest exponent x is to, in this case, x has no exponent so there can't be more than one solution, so we say that it has one solution. This way is a lot faster, but can be wrong if the equation has a false solution. This false solution happens when you solve the equation and get a value, but it doesn't work for the original equation. 

The second way is a lot faster, but the first way will always give you the right answer