Inequalities problem

joannabanana wrote:If –2<a<11 and 3<b<12, then which of the following is NOT true?
(A) 1<a+b<23
(B) -14<a-b<8
(C) -7<b-a<14
(D) 1<b+a<23
(E) -24<ab<132

I know what the answer is, but I don't quite get how to go about solving this.
Any tips on the quickest way to solve it?

I'm afraid it comes down to just going through each choice and plugging numbers into them. However, in this problem, a good starting point be to notice that choices A and D are identical, therefore the answer can't be A or D (there can't be two right answers : p) So you'll be left with only three options to evaluate which shouldn't take you more than two minutes to go through.

joannabanana wrote:If –2<a<11 and 3<b<12, then which of the following is NOT true?
(A) 1<a+b<23
(B) -14<a-b<8
(C) -7<b-a<14
(D) 1<b+a<23
(E) -24<ab<132

I know what the answer is, but I don't quite get how to go about solving this.
Any tips on the quickest way to solve it?

The way I solved this problem is to jot down the two positive equalities (given) and their corresponding negative inequalities and then simply add them up..

-2 < a < 11 (given) ---------------------------------------------> 1
-11< -a < 2 (corresponding -ve inequality) -----------------> 2

3 < b < 12 (given) -----------------------------------------------> 3
-12 < -b < -3 (corresponding -ve inequality) ---------------> 4

So now we have 4 inequalities to deal with..Just add them as per the options..

Options A & D are actually the same and the easiest. Simply add 1 & 3 above and you get
1< a+b < 23 . Hence this is correct.

Option B. Here you need a-b, hence add 1 and 4. You will get
-14 < a-b < 8. Matches and hence correct.

Option C. Here you need b-a, hence add 2 and 3. You will get
-8 < b-a < 14. This does not match and is also not a subset of the range given in option (-7 to 14). Hence this is the incorrect option and thus the answer for this question.

Option E. Here you need ab, hence multiply 1 & 3. You will get
-6< ab < 132. Since this is a subset of the range (-24 to 132), this option is also correct.

This took some time to type, but it didn't take much time to solve..

Not sure if this is the quickest approach but works for me!! How did you approach this question?

I really like your answer scorpion, thanks!

And I know what my problem was: for (C) or "b-a" I got 5 < b-a < 1
by doing:
3-(-2) < b-a < 12-11
I now know that I need to rewrite -2<a<11 to -11<-a<2 but I still don't completely get why I have to multiply the "a" inequality first instead of just subtracting it... Though as I'm writing this I think it's starting to make more sense, but maybe you can still shed some light

joannabanana wrote:I really like your answer scorpion, thanks!

And I know what my problem was: for (C) or "b-a" I got 5 < b-a < 1
by doing:
3-(-2) < b-a < 12-11
I now know that I need to rewrite -2<a<11 to -11<-a<2 but I still don't completely get why I have to multiply the "a" inequality first instead of just subtracting it... Though as I'm writing this I think it's starting to make more sense, but maybe you can still shed some light

No problem..

The thumb-rule for inequalities is that the inequalities signs (<,>,=) change when multiplied by -1.

Example: 1 < x < 5
Say x=2: 1 < 2 < 5 - Correct

Now if I want to rewrite the above inequality for -2 (-x), if I don't change the inequality sign, it would look like -
-1 < -2 < -5
This is illogical since we know that -5 is less than -2 which is in turn less than -1. Hence the inequality sign must be changed to make it sensible.
-1 > -2 > -5
or
-5 < -2 < -1 - This makes perfect sense.

In the b-a illustration here, you finally got 5 < b-a < 1, which doesn't make sense in this context.

If all else fails,
a. try to plug in numbers
b. visualize the inequality on the number line

In this case the 5 < b-a < 1 inequality represents a range of numbers on the number line which are all less than 1 or all greater than 5. Not to say that this cannot happen but in this context it is not correct. In situations where that is really the intent, the inequality would have to be written as two separate inequalities i.e. b-a < 1 AND b-a > 5

Great, thank you! I think I just need to be more careful and look at whether what I've written down, as you said, makes sense!