Why flip the sign in an inequality?
Why do we flip the inequality symbol when multiplying or dividing by a negative number?
Consider the following inequality:
-x < 4
We could multiply both sides by (-1), but we'd have to flip the sign:
(-1)(-x) > (-1)4
and we would end up with the solution:
x > -4.
____________
In my experience, few teachers say why this is true. They just teach it as a rule without giving any justification, because that's simpler. Here I'll give an explanation.
We'll consider what happens if we start with the basic inequality
-x < 4
and instead of multiplying by (-1), why don't we add x to both sides:
-x < 4
+x +x
and we end up with
0 < 4+x.
Now if we subtract 4 from both sides:
0 < 4+x
-4 -4
we end up with
-4 < x
or equivalently:
x > -4.
This is the same result we got from using the mysterious "multiply by negative and switch the sign rule".
Consider the following inequality:
-x < 4
We could multiply both sides by (-1), but we'd have to flip the sign:
(-1)(-x) > (-1)4
and we would end up with the solution:
x > -4.
____________
In my experience, few teachers say why this is true. They just teach it as a rule without giving any justification, because that's simpler. Here I'll give an explanation.
We'll consider what happens if we start with the basic inequality
-x < 4
and instead of multiplying by (-1), why don't we add x to both sides:
-x < 4
+x +x
and we end up with
0 < 4+x.
Now if we subtract 4 from both sides:
0 < 4+x
-4 -4
we end up with
-4 < x
or equivalently:
x > -4.
This is the same result we got from using the mysterious "multiply by negative and switch the sign rule".