Why must you flip the inequality symbol when you divide by a negative number?

because if you divide a positive by a negative, it would then be a negative, and to say that a negative is bigger than a positive is untrue so you must flip to make the equation true.

--"i won't be able to use the reason XYZ, because of the fact it fairly is obvious" yet all of technological expertise is per employing regulations while they're obvious. the exciting section being, "of direction" to tutor that something that seems obvious to many, seems to be incorrect...! --- Multiplying the inequality reverses the process the "arrow" of properly-order relation. a set of factors, in a properly-ordered set (the only form of instruments the place inequalities make experience), basically have 3 allowed states, relative to an different ingredient of the set. enable a and b be 2 factors in a properly-ordered set: then you definately could desire to have one and easily between right here circumstances: a = b a < b a > b This could desire to be genuine for ANY pair of factors interior the set. no count if it fairly is not, then the set isn't "properly-ordered" and an inequality must be meaningless. If we are saying, as an occasion, that a > b, then it means that there is a protracted way between a and b, and that this distance could be expressed via a non-0, beneficial cost, such that a = b + ok, the place ok is that non-0 beneficial cost. If we've 2 numbers with the different relation, as an occasion c < d, then we are able to apply yet another non-0 beneficial cost (enable's call it m) and say c = d - m returned to our a > b, corresponding to a = b + ok multiply the two facets via -a million -a = -b - ok this is permitted, because of the fact the equivalent sign isn't tormented via any multiplication, as long because of the fact the multiplication is carried out to the two facets. although, we at the instant are employing a minus register front of the non-0, beneficial cost ok. That could desire to be interpreted as -b being on the "different" area of -a -a < -b that's as though the detrimental multiplication had flipped the sign. What it fairly did is take the separation (ok) between the unique a and b, and adjusted its course while we went to their additive inverses -a and -b

I can only give you an intuitive answer, this is by no means a proof.

First: both division and multiplication are essentially the same operation, if you are dividing a number "a" by a number "b" you can be said to be multiplying "a" by "1/b": that is how we are getting our rules for dividing fractions etc. 6:3=2 is the same as 6*1/3= 6 *1/3=6/3=2 . From this follows that rules for dividing with negative numbers are the same as for multiplying with negative numbers.

Second: Any negative number can be written in the form of -1*a, so "multiplying/dividing both sides of an inequality by a negative number", is the same as "multiplying/dividing both sides of an inequality by a positive number and then multiplying both sides by -1"

Third: when you imagine your numbers on a number line, the positive numbers always "grow" to the right, and the negative ones always " get smaller" to the left. 1 is smaller than 2 but, -2 is smaller than -1 since each number on the left is smaller than the number immediately to the right of it.

Performing the operation 1+2 you will have travelled to 3 on the right hand side, and -1-2 will take you to -3, to the left. So when you perform operation on numbers with the same signs you "travel"/count on the number line in the same direction, either increasing or decreasing.

However what happens when you perform operations on numbers with mixed signs? Your "travel"/counting will change direction. So 5-3 will take you to 2 to the left of 5 and -3+5 will also take you to 2 by to the right of -3, the number that you have started with, you will get to the same result but from different directions on the number line.

This works similar not only with operations between numbers, but also with relations between them: (3<5)

so 3 is smaller than 5, but when you look at the number line -5 is smaller than -3 (-5<-3), so -3 is grater than -5 (-3>-5). What really happened here is that by multiplying both sides by -1 you have reversed the direction of the inequality.

This also happens when you are dividing by a negative number:

since 6>4 (divided by -2)

is 6>4 (divided by 2 and multiplied by -1)

(3>2)*-1

multiplying by -1 changes the direction of the inequality, since -3 is smaller than -2

so we get -3<-2