If equation a holds and equation b does not hold is it necessary for a+ b to not hold?

Question

I've been using this trick a lot lately, I can't seem to prove why it works. E.d. (using an inequality here, but point still remains)

:

$$|x-a| \ge |x| - |a| \tag {1}$$

add $$|x|+|a|\ge |x-a|$$

$$|a|\ge|a|$$

Therefore, $(1)$ is true.

Answer 1

You did

$$\text {unknown inequality}+\text{true inequality} = \text{true inequality}$$

and concluded that the unknown inequality is true. This does not work in general. Take

Unknown inequality: $2>3$

True Inequality: $5>1$

If you add these up you get a true inequality, $7>4$. But as you see this doesn't mean that the unknown inequality must've been true.

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Now with equations it's a different matter.

Suppose you have

Unknown equation:$a=b$

True equation: $c=d$

True equation: $a+c=b+d$

To see that the unknown equation is true, just do

$$\text{second true equation} - \text{first true equation}$$

You know that both equations are true so the result of this must be true. But the result is $a=b$, the unknown equation!

Answer 2

Just $|x-a|+|a|\geq|x-a+a|=|x|$.

By your reasoning we can prove that $4>5$.

Indeed, since $3>1$ we "obtain": $$3+4>1+5,$$ which is true.

"Hence", $4>5$.