could someone help me how to approach this problem

[aspirant1]

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<152

OA C

NOTE : If i go by plug in values...its going on for ever!; then i tried to solve the inequalities i.e -6,33,-24,132 what's the next step! I am trying to understand how to solve/approach this problem.

Thanks

[papgust]

This is how i solved.

**************************************************
The method is if you have the inequality in the format
w<a<x
y<b<z

then the following are TRUE,
(1) w+y < a+b < x+z
w+z < a+b < x+y

(2) w-y < a-b < x-z
w-z < a-b < x-y

(3) w*y < a*b < x*z
w*z < a*b < x*y

***************************************************
Apply the method to this problem,

-2<a<11
3<b<12

(1) 1<a+b<23 ........ (The inequality is present in choice #1. It is TRUE. Eliminate #1)
10<a+b<14 ....... (The inequality is NOT present in the choices. So, Ignore)

Choice #4 is exactly the same as choice #4. So, this is also TRUE. Eliminate #4.

(2) -5<a-b<-1......... (The inequality is NOT present in the choices. So, Ignore)
-14<a-b<8 .......... (The inequality is present in choice #2. It is TRUE. Eliminate #2)

(3) -6<ab<132........ (The inequality is NOT present in the choices. So, Ignore)
-24<ab<33 .......... (If ab < 33, then ab is also < 152. So it is TRUE. Eliminate #5)

The only option left is C.

[sadullaevd]

great explanation,

thanx

[2009wish]

@ papgust,

How can we say that if ab <33 then ab can also be less than 152?

Also, b-a .... we get -8<b-a<14 ......isnt it much closer to option C rather than 33-152?

[papgust]

Good catch. You have a point here. I didn't calculate b-a when i posted the solution.

Maybe experts could step in.

[Stuart@KaplanGMAT]

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<152

OA C

NOTE : If i go by plug in values...its going on for ever!; then i tried to solve the inequalities i.e -6,33,-24,132 what's the next step! I am trying to understand how to solve/approach this problem.

Thanks

Honestly, I'm not at all sure that knowing the rules for every possible iteration is faster than brute force; I'd have just jumped right into the choices.

It's a "which of the following" question, which means that the answer is slightly more likely to be near the bottom (based on historical GMAT stats), so, let's start with (e) and work our way up:

e) -24<ab<152

Only a can be negative, so let's pick the smallest possible a and the biggest possible b to get our smallest value for ab:

(-2)(12) = -24

To maximize ab, let's pick the two biggest values:

(11)(12) = 132

So, we get:

-24 < ab < 132

As papgust points out, if ab < 132 it's certainly less than 152, so (e) must be true: eliminate it.

d) 1<b+a<23

the minimum value of b+a uses the minimum value of each:

(-2) + (3) = 1

if we add the maximum value of both:

11 + 12 = 23

So, we know that:

1 < a + b < 23. As papgust also right points out, (A) and (D) are identical, so eliminate both of them.

c) -7 < b - a < 14

The minimum value of b - a occurs when we minimize b and maximize a:

3 - 11 = -8

We can already see that b-a could be less than or equal to -7, so (c) is NOT a must be true: choose C.

Now, it takes awhile to say all that out loud or to type it up, but if you know how to test each choice you should be able to answer this question by brute force in well under 2 minutes.

* * *

As an aside, if this question were properly worded, it would have read "which of the following is not necessarily true" instead of just "which of the following is not true"; with it's current wording, there's no correct answer, since it's possible that b-a falls in the range in (c), it's just not a sure thing.