Show that if ab, then (b^-1)(a^-1), provided that a and b have the same sign. What happens if a0 and b

Question

Show that if a>b, then (b^-1)>(a^-1), provided that a and b have the same sign. What happens if a>0 and b<0?

anonymous · Answer

well:$$a>b$$dividing by each variable we get:$$a/(ab)>b/(ab)\1/b>1/a$$note the division is safe and we don't need to worry about the direction of our inequality flipping since $a,b$ are of the same sign and thus $a,b$ is nonnegative

anonymous · Answer

oops, $ab$ is nonnegative (their product). this makes sense since a positive times a positive yields a positive, a negative times a negative yields positive, and 0 times anything yields 0!

anonymous · Answer

now, if $a>0,b<0$ we no longer have the same sign and thus $ab$ is not necessarily nonnegative; in fact, we actually know $ab<0$ so when we try the same approach above dividing by $ab$ necessitates 'flipping' the direction of the inequality:$$a>b\a/(ab)

anonymous · Answer

Thanks! I definitely wanted to make sure that 1/b was the inverse of b. I think for some reason I got it in my head that that wasn't true after I worked on inverse functions- silly me XD So thanks again!