why does the direction of the sign of an inequality change when you divide by a negative number?

Question

⎟1-2x⎟> 3

Answer 1

The reason why the sign changes is as answered above. Since multiplying or dividing by a negative number reverses the signs of both negative and positive numbers, each number will be equally reversed in magnitude.

Initially you requested the inequality be solved.

⎟1-2x⎟> 3
This is a special case that contains absolute value, thus in order to solve, it must expressed as a set that includes the negative and positive range:

-3 > 1-2x > 3

subtract 1 from both sides:

-3 - 1 > -2x > 3 - 1
-4 > -2x > 2

divide by -2 and reverse the signs

2 < x < -1

By convention, this is expressed with the negative number on the left :

-1 > x > 2

This is the set of numbers that satisfy the inequality. This means that only the small range of numbers between -1 and 2 do NOT satisfy the inequality.

Answer 2

If you multiply (or divide) any number by a negative number that number will change signs. That is if the number is already negative, the negative sign will cancel. If there is no sign present, the sign will now be present.

So clearly, if the number was negative it will become positive, and vice versa. The quantity crosses over zero on the number line.

So in an equation, each side (left and right) will cross zero. This means whichever one was greater is now the lesser of the two. And vice versa.

Pick a couple of numbers on the real number line, say 6>-2. Divide by -2. Now -3<1.

I hope that helps

Answer 3

While a big positive number is bigger than a small positive number, a big negative number is technically smaller than a small negative number.
10>1
-10<-1
When you multiply or divide by a negative number, you change the signs on both sides of the equation, requiring you to switch the inequality sign.

Answer 4

Logic.

Answer 5

Sorry I cant remeber why, it does though.