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I get myself confused sometimes when to use => or =

= x + y = x + 5 = x = -5


=> x + y => x + 5 => x = -5
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See also "Stream-of-consciousness equalities and implications" section of Common Errors in College Math: – Theta33 Mar 14 '11 at 6:41
up vote 15 down vote accepted

Neither of those are right. It should be: $$x + y = x + 5 \implies x = -5.$$ (Though I have to say, I'm not exactly sure how $x = -5$ directly follows from $x + y = x + 5$ to begin with.)

Let's first distinguish between expressions and equations. Expressions are things like $x + 2$ or $37$ or $x^2 - 2x$. On the other hand, an equation is an equality between two expressions, such as $x + 2 = 3$ or $x = 1$ or $8 + 9 = 17$ and so on. So, an equality says that two expressions are the same, and we use the equals sign $=$.

Note that an equality is making a logical claim (true or false), whereas an expression is not. For example, $6 = 6$ is a true equality, whereas $1 = 2$ is a false equality, whereas $17$ is just an expression (not true or false).

Now, the implies sign $\implies$ is used to show that one equation can be logically deduced from another. So, the implication $$x + 2 = 3 \implies x = 1.$$ means that from the equation $x + 2 = 3$, we can conclude that $x = 1$.

Note that it does not make sense to use the implication arrow between expressions. For example $x + y \implies x = -5$ does not make any sense, because the $x + y$ is an expression. After all, what is the truth value of $x+ y$ anyway? There isn't one.

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The equal sign $"="$ should be used when you have two quantities which are equal. For example:

$$x^2 + 3x - 4 = (x+4)(x-1).$$

The $"\Rightarrow"$ should be used when relating two statements. For example:

$$ \text{Today is Tuesday} \Rightarrow \text{Tomorrow is Wednesday}.$$

It is okay to mix them, say for instance you are solving an equation like the following:

$$ 3x + 4 = 10 \Rightarrow 3x = 6 \Rightarrow x = 2.$$

The quantities $3x + 4$ and $10$ are equal. However, the statement $3x + 4 = 10$ implies that $3x = 6$.

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I think that equal sign is used for showing equality between first and second step but implies that is used for getting into the next step.

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