Question related to compound inequality

[PavanK217]

Hi Folks,

I am currently reviewing the material from Manhattan GMAT. This is related to compound inequalities.

Here is the question:

If x>8, x<17, x+5<19, what is the range of possible values of x?

Solution:

Solving the above individual inequalities and making the equalities to point in the same direction we get these - 8<x, x<17, x<14

I see two possibilities: 8<x<14 OR 8<x<17

But the book prefers 8<x<14 as this is a tighter range than 8<x<17

Let's say x=15. This would not fall in the inequality 8<x<14 but x=15 still satisfies the individual inequality x<17 (given in the question).

Is it correct to conclude 8<x<17 would be a better option than 8<x<14 OR is there something I am missing?

[Sage Pearce-Higgins]

Please provide the source and location of examples you cite. The book is correct on this issue, which touches the wider topic of how mathematical statements work. When a problem cites some mathematical statements as facts, then they are true. In this problem, we know that x>8, x<17, and x+5<19. As per your good algebra, we know that - 8<x, x<17, x<14. Now, rather than thinking of these as abstract mathematical statements, consider them as 'the number of dollars that Victoria has'. We know that this number is bigger than -8, and we know that it's smaller than 17, and smaller than 14. Can it be 15? No. That number doesn't agree with our information. Sure, it's smaller than 17, but we also know that it should be smaller than 14. At this point you're probably thinking 'the fact that x is smaller than 17 isn't really telling me anything useful' and you'd be right. In the light of knowing that x < 14, the fact (note: it is still a fact that is true) that x < 17 is basically useless.