vkb16 wrote:
btw, can we flip the inequality sign if we flip the fraction? (numerator = denominator, and vice versa)
for example, if 1/2 > 1/5, 2<5
You should be careful doing that. It's perfectly correct if you know that all of your numbers are positive, but it can be incorrect if negatives are involved. For example, (-1/2) < 1/2, and if you flip the fractions, -2 is still less than 2.
When you 'flip the fraction' in an inequality, you're really multiplying and dividing on both sides. For example, say a, b, x and y are all positive, and that
a/b < x/y
Since all our letters are positive, we can multiply and divide on both sides without needing to reverse the inequality:
a < bx/y [multiply by b on both sides]
ay < bx [multiply by y on both sides]
y < bx/a [divide by a on both sides]
y/x < b/a [divide by x on both sides]
Which is why, when all your numbers or letters are positive, you can flip fractions on both sides of an inequality as long as you also flip the inequality. Of course, if one or more of our letters is negative, at some point in the above, we might have needed to reverse the inequality, so this 'rule' would no longer apply. Rather than learn different 'rules' here for different combinations of negative and positive letters, you can always do as I did above - multiply and divide on both sides - and if at each stage you consider whether you might have multiplied or divided by a negative, you'll never make a mistake.
