Inequalities combination

[sshin]

Hi,

I am having trouble grasping the concept of combining inequalities. To list an example problem that was in the All the Quant Guide (#6, page 241):

If a>7, a+4>13, and 2a<30, which of the following must be true...

The possibilities either come down to 7<a<15 or 9<a<15. The explanation for why 9<a<15 is correct is: "If 7 and 9 are both less than a, the limiting factor is the larger value, 9. (If a is greater than both 7 and 9, then overall a is greater than 9)." Is it possible to explain this a bit further? Isn't 7 more "limiting" than 9?

There was a similar example earlier in the chapter: "If x>8, x<17, and x+5<19, what is the range of possible values for x?" I understood why x<14 is more limiting than x<17 but I don't know how the "limiting" works for the prior example.

Also, does the rule change when we are dealing with negative numbers? If there's an example to that, that'd be super helpful as well.

Thank you

[Sage Pearce-Higgins]

I agree that this is a confusing area of Math. My approach is to turn these mathematical sentences into words to understand them better. To take an example, if I told you that 'Alex is older than 7', you can easily see that Alex could be 8, 9, 10, etc. Now, suppose you have two facts: (1) Alex is older than 7 and (2) Alex is older than 9. Note really carefully that these two facts are true; don't try to argue against them. Now think: could Alex be 8? Pause and consider.

Alex couldn't be 8, because one of the facts says that he's older than 9. No worries that the other fact says that he's older than 7. That's true, but it's actually pretty useless as the second fact tells us more specific information. Here, that first fact is just a bit of a distraction.

When it comes to negative numbers, the same process works, but you need to avoid the confusion that negative numbers can bring about. Which number is bigger -5 or -4 ? Again, think of a real situation: who's richer, someone who owes $4 or someone who owes $5? I know that $5 is a bigger debt, but the person who 'only' owes $4 is richer, so that -4 > -5.

[sshin]

Hi Sage, thank you for that explanation - the real world example makes it so much easier to grasp the concept! So it sounds like the mechanism is really that 1) you want to take the biggest number on the left side of the "<" inequality sign and 2) take the smallest number on the right side of the ">" inequality sign. Is that a fair of way of thinking about it?

[Sage Pearce-Higgins]

I'm glad to hear that my explanation makes sense. I often find that putting something in a more familiar context can help us understand it.

As for your comment:

1) you want to take the biggest number on the left side of the "<" inequality sign and 2) take the smallest number on the right side of the ">" inequality sign

This doesn't make sense to me. There's just a single number (or letter) on each side of a "<" or ">" symbol, so saying that you can take the biggest number suggests that you're misunderstanding things. Also, the phrase 'you want' is pretty vague - it depends what you're trying to do. I suggest that, instead of thinking in such terms, you focus on understanding an inequality as a mathematical sentence. The statement x < 5 simply means "x is a number less than 5". Considering a few quick examples might help, so that thinking 'right, x could be 4, or 3, or 4.5, or -7, etc.' might help us apply this statement to various situations.