How best to explain why, when you multiply an inequality by a -ve number, you must swap the inequality sign?

Both methods will work well. Perhaps this might help:

Say if we have 2 numbers a and b, and a>b. Then, adding -a-b to both sides, we get -b>-a. Addition preserves the direction of the inequality.

if you multiply by a negative you are changing the VALUE of both sides of the inequality..

when you change the sign of both sides of a equation there is no consequence it is still an equation..

but when you change the sign of a inequality it must be flipped because of the nature of inequalities..

a number is greater than another number if its to the right of that number on the number line

if I were to change the signs of both sides of an inequality the larger number now becomes the SMALLER negatively number..

this will make the inequality untrue invalidating the form

if i have the formula

10 < 22

you can see this is true

if i multiply by -1 both sides

-10 < -22 is untrue

because the larger number before now becomes the smaller number

this is the same with - numbers

-10 < -5

multiply by -1 makes this invalid

10 < 5

larger number -5 becomes the smaller number

this works even if one is one sign and the other is the other

-5 < 10

time by -1

5 < -10

again the larger number becomes the smaller number

that is the reason why the inequality is flipped

you can look at this as translation of the points on the number line..

I believed D G and Emily pretty much explains it all. There's another case where you need to you swap the inequality sign. It's when you raised both sides of the inequality to a negative power (such that both are still defined).

e.g

3 < 4

3^(-2) > 4^(-2)

Here is how I teach it..........

You definitely agree (I ask the students) that........

8>-2....................they agree

Divide both sides by 2

4>-1..................(I ask) Do you agree with this statement.......they agree

THEN.......again

You definitely agree (I ask the students) that........

8>-2....................they agree

Divide both sides by.....-. 2

-4>+1..................(I ask) Do you agree with this statement.......they don't agree

Then I say well what if we changed the direction (or Order) of the ....>

-4<+1.................they agree

Then i tell them the rule about mult/dividing an inequality by a negative number you MUST REVERSE THE Inequality sign