Question

What happens to the inequality when you multiply or divide by a negative number

Answer 1

Explanation:

When you multiply or divide both sides of an inequality by a negative, you swap the inequality. You had less than or equal? Now it'll be greater than or equal.

Answer 2

flip the sign

Explanation:

You just flip the sign...

for example:
`-x < 2\implies x > -2`

But why is this happening? So let's just analyze the initial inequality we started with:
`-x < 2`

As you can see here, the x values are negated, or in other words they become negative. So even though -5 is less than 2, it becomes -(-5) which cancels the negative and it becomes 5, this of course makes the inequality false.

The opposite occurs with positive values, the positive value becomes negative, thus any positive value will satisfy this inequality, since it becomes a negative number.

For example 10 is not less than 2, but the negative sign will make it -10, thus making the inequality true.

Even though negative values will cancel out, there are still some negative values that satisfy this equation. Since -1 becomes 1, it satisfies this equation, but once it becomes less than or equal to -2, it becomes false. This is because it becomes a positive value. Thus the equation really just means x must be greater than -2.

I could've used something more abstract, but I wanted to give a specific example so it might be a bit easier to understand.