# Is using a $\Rightarrow$ to separate simplification steps valid?

Say I have function $y -2 = -4(x + 1)$. Is it a valid use of $\Rightarrow$ to, in the course of simplification, do this:

$y -2 = -4(x + 1) \Rightarrow y = -4x -2$?

Both sides of the equation are equal, and from what I understand of that symbol this is appropriate, but I'm not sure.

For all of high school I did something similar to this, except I just used $\rightarrow$ between the steps, especially in Algebra, to keep the actual simplifications separate, and never had any issues with any teachers doing this. I felt it kept my work cleaner, but my Calc professor marks us down for doing that, and I'd prefer to be able to separate steps out a bit more than

$y -2 = -4(x + 1) = y = -4x -2$

• I'm also not sure what exactly to tag this with, it's my first time posting on Mathematics and none of the tags that really applied that I could think of existed. – RPiAwesomeness Sep 14 '17 at 3:11
• Note that in a chain of equals signs $a=b=c$ implies that $a=b$ that $b=c$ and by transitivity of the equals sign that $a=c$ as well. Your final line that you wrote reads "$y-2=-4(x+1)=y=-4x-2$" and would have implied that $y-2=y$ which would imply that $0=2$ which is not true if we were talking about real numbers. Use equals signs only for equality and not for "continuing to next step" markers. Use extended blank space, or arrows, or something else other than an equals sign for that. – JMoravitz Sep 14 '17 at 3:15

I'm not entirely sure what the question is. In logic, the $\implies$ symbol means "implies." So it is certainly true that $y-2=x\implies y=x+2$, for example. In your second example, it makes no sense to write

$$y -2 = -4(x + 1) = y = -4x -2$$

since you have just written $y-2=y$ and thus $-2=0$.

I'm not sure why any instructor would mark down for using $\implies$, unless she specifically asked you to use english words. Maybe you should check in with them.

Yes, that should work fine: the $\Rightarrow$ means logical implication, and the statement $y-2=-4(x+1)$ does indeed logically imply that $y=-4x-2$

What you do at the end, though:

$y-2=-4(x+1)=y=-4x-2$

is certainly not correct, because you end up saying that $y-2=y$

• Maybe $-2 = 0$? – Xander Henderson Sep 14 '17 at 3:29