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Check this one out:

Solve

-3x is less than or equal to 6, ditch the -3 by dividing both sides by -3 which gives x is less than or equal to -2

It looks ok ... But, is it?

number line showing x is less than or equal to -2

This means that x can be -2 or any other number less than -2.

Let's check!

         -3x is less than or equal to 6

when x = -2, -3 (-2) is less than or equal to 6, which gives 6 is less than or equal to 6.  Yep - that works.

when x = -4, -3 ( -4 ) is less than or equal to 6, which gives 12 is less than or equal to 6.  No way, dude!

It didn't work.  Wazzup with that?

Here's the freaky thing:

When you divide (or multiply) by a negative number, you mess up the inequality sign!