Question: If -2 < a < 11 and 3 < b < 12, then which of the following is NOT always true?

1. 1 < a + b < 23
2. -14 < a - b < 8
3. -7 < b - a < 14
4. 1 < b + a < 23
5. -24 < a b < 132

Correct Answer: C
Solution and Explanation:
Approach Solution 1:
a+b…
For range, take max values of both for upper end and min values for lower end.
upper end : 11+12=23 & lower end : -2+3=1.... So  1 < a + b < 11
A & D are correct

a-b…
For range, take max value of a and min value of b for upper end and min value of a and max value of b for lower end.
upper end : 11-3=8 & lower end : -2-12=-14.... So  -14 < a – b < 8

B is correct

b-a.
For range, take min value of a and max value of b for upper end and max value of a and min value of b for lower end.
upper end : 12-(-2)=14 & lower end : 3-11=-8.... So -8 < b – a < 14 but given is  -7 < b – a < 14
C is Not correct

ab…
For range, take max value of a and max value of b for upper end and min value of a and max value of b for lower end.
upper end : 11*12=132 & lower end : -2*12=-24... So  -24 < ab < 132
E is correct ..

Approach Solution 2:
We have a range of a: -2 < a < 11. This means range of -a: -11 < -a < 2
We have a range of b: 3 < b < 12. This means range of -b: -12 < -b < -3

Now, the range of (a+b) or (b+a): (-2+3) < a+b < (11+12) Or 1 < a+b < 23.
So A and D are true.

Range of a-b is range of a+(-b): (-2-12) < a-b < (11-3) Or -14 < a-b < 8
So B is also true

Range of b-a is range of b+(-a): (3-11) < b-a < (12+2) Or -8 < b-a < 14
As we can see, C is not always true

Approach Solution 3:
If we add the two inequalities together, we have:

1 < a + b < 23

Thus, A is true and since a + b = b + a, D is also true.

Multiplying the second inequality by -1, we have 3 > -b > -12 or -12 < -b < 3. Now, adding the latter to the first inequality, we have:

–14 < a – b < 8

So B is true.

Similarly, multiplying the first inequality by -1, we have 2 > -a > -11 or -11 < -a < 2. Now. adding the latter to the second inequality, we have:

–8 < b – a < 14

So C is NOT true since (b - a) could be -7.5, which does not fall into -7 < b - a < 14.

“If -2 < a < 11 and 3 < b < 12, then which of the following is NOT always true?”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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