The general rule is that if you perform an operation to one side of an equation you must then also perform the *same* operation to the other side. When I was in grade school we were taught this with the silly mnemonic device "Whatever you do to grandma, you have to do to Grandpa too" :roll: In this instance, you've performed two operations - "Adding x and then [subtracting] 1/2." Both times you performed an identical operation to both sides of the equation, so you're fine as far as that goes.

But if, for whatever reason, you still find yourself doubting, another approach is to note that two equations are considered equivalent if and only if they have the same solution set. So ask yourself, how many solutions are there to \(\displaystyle 5x = \dfrac{3}{2}\)? And what are they? If you plug the solution(s) into the other equation \(\displaystyle \dfrac{1}{2} + 4x = -x + 2\), what do you find? Do the solution(s) to \(\displaystyle 5x = \dfrac{3}{2}\) also satisfy \(\displaystyle \dfrac{1}{2} + 4x = -x + 2\)? Lastly, how many solutions are there to \(\displaystyle \dfrac{1}{2} + 4x = -x + 2\)? Are there any other solutions that haven't already been accounted for? Based on all of this, what can you conclude about whether the equations are equivalent?