False answer from an equation

Question

If you have the equation $$ \left(\frac{1}{x}\right) + 1 = 0, $$ and you solve it like this $$ \left(\frac{1}{x}\right) + 1 = 0 $$ $$ \left(\frac{1}{x}\right) = -1 $$ $$ -x = 1 $$ $$ x = -1, $$ everything's good. But if you do it like this $$ \left(\frac{1}{x}\right) + 1 = 0 $$ $$ \left(\frac{1}{x}\right) = -1 $$ multiply both sides by x $$ x = -x $$ divide by x $$ 1 = -1, $$ you get nonsense. What is going on? What am I missing here.

Answer 1

You've made a computational error.

$$\frac{1}{x}=-1$$

multiply both sides by $x$ gives

$$1=-x$$

Math is still safe to use.

Answer 2

There are two things wrong. The first is that you have multiplied $\frac{1}{x}$ by $x$ and got $x$ instead of $1$.

The second is that you have divided by a variable without first determining that the variable is not equal to zero.

Answer 3

$\frac1x*x=1 $ not $x$.You have got it wrong.