Question on an inequality rule

Question

Hello my GMAT friends, I am aware of the rule that you should never subtract or divide inequalities. However, I would be so appreciative if someone can provide an example as to what this looks like. For instance, I am assuming the rule below means that the following examples I made-up would not be allowed below: Example 1... 7 > 2x > 2 - 3 > x > 1 ---------------- = 4 > x > 1 Example 2... 10 > 10x^2 > 5 ÷ 5 > 5x > 5 = 2 > 2x > 1 Example 3... What about simplifying through division? 20 > 2x > 2 --> divided by 2 = 10 > x > 1 In other words, is the rule to not divide one inequality into another and to not subtract an inequality from an inequality? Thank you in advance. avigutman hoping that you will be the one to show me the light on this after some crickets

Thank you for your time.

avigutman · Answer

woohoo921  I suggest that you draw the inequality on a number line and think through the arithmetic operations that you're performing. Split those operations into two categories: Additive vs Multiplicative.

Here we know the relative order of three terms on the number line:
_____2______2x______7_______
And we also know the relative order of these three terms:
_____1______x______3_____
From here, it might help to think of the first three terms as the weights of three people who are about to go on a diet, and the second set of three terms as the weights that each of them managed to lose. So, in this example, the heaviest person, call him A, weighed 7, and lost 3, so his weight at the end was 4. The second person, call him B, weighed 2x and lost x, so ended up weighing x. The last person, call him C, weighed 2 and lost 1 so he ended up weighing 1.
Now, consider this thought experiment: If the heaviest person lost the most weight, and the lightest person lost the least weight, are we able to infer who's heavier at the end? Think about this for a bit. What do you think? Why or why not?

Another way to think about this: we know that adding inequalities is okay (if the heaviest person gains the most weight, and the lighter person gains the least weight, obviously the heaviest person is still heavier than the lightest person at the end). When you attempt to subtract one inequality from another, you're effectively multiplying that inequality by (-1) and then adding the inequalities. Adding them is okay, of course. But, can we multiply an inequality by (-1)? No. Why not? Because that would throw the terms to the other side of zero, and we'd get the mirror image of the original inequality.
 
Here we have a multiplicative operation (dividing is conceptually the same is multiplying; for example, multiplying by (3/2) is the same as dividing by (2/3). Now, multiplicative operations in general are very problematic with inequalities, because the terms can easily get thrown over to the other side of zero and give us their mirror image. Let's assume for the sake of argument that we're only dealing with positive numbers in this example. Okay, so what are you doing here? Dividing the largest number by the largest number and the smallest number by the smallest number, and wondering whether the resulting ratios will be arranged in a predictable manner? Remember, dividing by a larger number makes you smaller, and vice versa. So, if we were told that we're dealing with just positive numbers, we  could  make the following inference:
If 5x>3y>0 and a>b>0, then 5x/b > 3y/a
 
Here you're just reducing a given inequality by a factor of 2, which is the same as expanding the inequality by a factor of (1/2). Nothing wrong with that. On the number line, you're cutting all the distances of all the terms from zero in half. That doesn't change their relative order. It's like zooming in or out on a map. The distances among the cities change accordingly, but the general placement stays the same (New York is still east of California, and Canada is still north of the USA).

Question on an inequality rule

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