# Why does the inequality sign change when both sides are multiplied or divided by a negative number?

Update:

algebra question for which i am having a lot of difficulties with.

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• Anonymous

Because positive and negative numbers are symmetric about zero.

Consider the action of multiplication by a negative number on two distinct points on the number line, and their distances relative to zero.

Multiplying by a negative number is equivalent to multiplying by the absolute value of that number followed by multiplication by -1.

Multiplying by a positive number merely stretches each of the distances relative to zero, and the direction of the inequality is preserved.

Multiplying by -1 inverts the sense of the distance relative to zero, reflecting the configuration of the points over the y-axis at x=0. As a result, the direction of the inequality must also be inverted in order to preserve the correct sense of the magnitude relationship between the distances.

[If dividing is considered multiplication by the reciprocal, the same argument used for multiplication can be applied for division.]

Because negative numbers with larger magnitudes are smaller than negative numbers with smaller magnitudes. If you didn't change the sign, it wouldn't be true anymore.

Maybe I could come up with some informal justification here. Suppose a < b. and -a < -b. Add those two inequalities together and you get 0 < 0. That's not true, so one of those has to be wrong.

a <= b is trickier, I guess. Suppose a <= b and -a <= -b. That's trickier . . . Start by dividing -a <= -b by b. Then -a/b <= 1 (remember, you're not switching the equality signs, so it doesn't matter if b or a are negative). Divide the first by a. Then 1 <= b/a. So -a/b <= b/a. -a^2 <= b^2. -a <= b. Now all you have to do is pick an a and b that make this statement false. Something like a = -4 and b = -3. The weakness of this is that a and b can't be zero, so you need to account for that with more explanation.

Peace!

Think of it as a mirror. The multiplication turns a number to it's negative equivalent.

If you have 5 > 4 and multiply it by -1 it will be -5 > -4 which is not correct. That's why you have to change the sign

-5 < -4 = Correct!

• luzell
Lv 4
3 years ago

cope with inequality warning signs only like equivalent warning signs different than, whilst multiplying or dividing the two aspects by ability of a destructive quantity, replace the inequality warning signs. in case you do no longer, the respond won't come out staggering. that's only how that's.

Aren't you surprised that - 24 < -4?

When you multiply (or divide) by a negative number, it becomes negative (if it was positive before) and among negative numbers "the more the less"

And vice versa if it was negative it becomes positive where everything becomes conversely (upside-down)

• Anonymous

I was having that problem to long time ago. What I did was to tell my self that everytime when you multiply or divide just change the sign to an opposite direction. This includes the equal sign(=), it does change the sign but you can't see it....

Multiplication by a negative is an abstraction that is visualized best by imagining a reflection.

If you have a tree top and an airplane flying above it, and you look at the reflection, you have an airplane flying below the the treetop. What was "above" (greater than) becomes "below" (less than) in the reflection.

Let x = a positive number

Let y = a positive number

let's say:

x > y

add -2x to both sides (which, in effect is like multiplying the left by -1)

x - 2x > y - 2x

-x > y - 2x

the equation is now invalid (graph it and you'll see).

However, if you switch the sign the equation becomes valid.

-x < y - 2x

• Amit Y
Lv 5

Be a and b two variables such that a < b

Then

a - b < 0

b - a > 0

Subtract b from the second inequality and get

-a > -b