Problem on inequalities

[krizia001]

On page 293 of the princeton review manual, question num. 2 there's a problem about inequalities that is driving me crazy because I think there should be and minus or equal sign not just the minus. I've tried to do it a lot of times, and it's an easy excercise, it's just that I think the signs are mistaken.

Problem:

If -2<a<11 and 3<b<12, then which of the following is not true?

___ 1<a+b<23

___-14<a-b<8

___-7<b-a<14

___1<b+a<23

___-24<ab<132


This is it! Thank u

[sanju09]

On page 293 of the princeton review manual, question num. 2 there's a problem about inequalities that is driving me crazy because I think there should be and minus or equal sign not just the minus. I've tried to do it a lot of times, and it's an easy excercise, it's just that I think the signs are mistaken.

Problem:

If -2<a<11 and 3<b<12, then which of the following is not true?

___ 1<a+b<23

___-14<a-b<8

___-7<b-a<14

___1<b+a<23

___-24<ab<132


This is it! Thank u

This is true that we can add/subtract similar inequalities

-2 < a < 11

3 < b < 12

Added

1 < a + b < 23, A true D true; eliminated

-2 < a < 11

3 < b < 12

Subtracted

-14 < a - b < 8; B true, eliminated

3 < b < 12

-2 < a < 11

Subtracted

-8 < b - a < 14; [spoiler]C needs repairs

C[/spoiler]

[limestone]

Just to continue with sanju09's explanation

We have proved that -8 < b - a < 14,

In the third statement: -7 < b - a < 14. The minimum value of (b - a) is -8 which is smaller than -7, so if (b-a) is so near to -8 (for example -7.9), it will be smaller than -7. This statement is not true.

For the last statement, the product ab must fall in the range between the product of max positive a & max positive b ( 11 x 12 = 132) and the product of min negative a & max positive b ( -2 x 12 = -24)

Why not a product of min negative a & min positive b? Let's try: -2 x 3 = -6 which is bigger than -24. So this statement is true.

So in my opinion, only the third statement is not true.

[krizia001]

You guys explained it verry well. The only question that I have is: Should I use the numbers although they are not included < >? For example -2 and 12 are not included.


Thank u again.

[nithi_mystics]

This is true that we can add/subtract similar inequalities

-2 < a < 11

3 < b < 12

Added

1 < a + b < 23, A true D true; eliminated

I think this is incorrect.

-2 < a < 11 ==> a can have values -1,1,0...10
3 < b < 12 ===> b can have values 4,5,6...11

So a+b can have the values 3,4...22 ==> 2 < a+b < 23

a+b cannot have the value 2 and hence 1 < a + b < 23 is incorrect.

I believe the question should have been "which of the following is true" The answer would be C.

[CompBanker]

This is true that we can add/subtract similar inequalities

-2 < a < 11

3 < b < 12

Added

1 < a + b < 23, A true D true; eliminated

I think this is incorrect.

-2 < a < 11 ==> a can have values -1,1,0...10
3 < b < 12 ===> b can have values 4,5,6...11

So a+b can have the values 3,4...22 ==> 2 < a+b < 23

a+b cannot have the value 2 and hence 1 < a + b < 23 is incorrect.

I believe the question should have been "which of the following is true" The answer would be C.

nithi_mystics: The question does not state that a or b are integers. "a" could be -1.9999 and "b" could be 3.00001. Adding the two together would make a+b>1, albeit ever so slightly.

[nithi_mystics]

My bad!!

This is true that we can add/subtract similar inequalities

-2 < a < 11

3 < b < 12

Added

1 < a + b < 23, A true D true; eliminated

I think this is incorrect.

-2 < a < 11 ==> a can have values -1,1,0...10
3 < b < 12 ===> b can have values 4,5,6...11

So a+b can have the values 3,4...22 ==> 2 < a+b < 23

a+b cannot have the value 2 and hence 1 < a + b < 23 is incorrect.

I believe the question should have been "which of the following is true" The answer would be C.

nithi_mystics: The question does not state that a or b are integers. "a" could be -1.9999 and "b" could be 3.00001. Adding the two together would make a+b>1, albeit ever so slightly.

[BlindVision]

-2 < a < 11

3 < b < 12

Subtracted

-14 < a - b < 8; B true, eliminated

Can you please show me how you got that answer by subtraction? I thought we can only subtract in the same direction of the < for both equations. I did not realize we can switch the orders around such that (-2 < a < 11) - (12 > b > 3). Please let me know what I'm missing...

[Ian Stewart]

Two things here:

• you should never, ever, subtract one inequality from another. You can add two inequalities which face the same way, but you can't subtract them (try subtracting the inequality 10 > 8 from the inequality 10 > 9 and you'll see that the result, 0 > 1, is complete nonsense). We aren't subtracting inequalities at all here. If we want to get information about a-b, say, we can multiply this inequality:
  
  3 < b < 12
  
  by -1. Since we're multiplying by a negative, we need to reverse the inequalities, so we find that
  
  -12 < -b < -3
  
  We can now add this to the inequality:
  
  -2 < a < 11
  
  to find that -14 < a-b < 8. If instead you want to learn about b-a, you can multiply the inequality -2 < a < 11 by -1, reversing the inequalities when you do, to get -11 < -a < 2, and then add this to the inequality 3 < b < 12. You'll find that -8 < b-a < 14.
• It's clear that C is the intended answer to this question, but if the wording of the question has been correctly transcribed in the original post, then this answer actually makes no logical sense. The question asks which inequality is "not true". Well, we know with certainty that -8 < b - a < 14; is the inequality -7 < b - a < 14 then 'not true'? We can't say; it might be true (b-a might be equal to 3, say) or it might be false (b - a might be equal to -7.5). To make logical sense, the question needs to ask 'Which of the following is not necessarily true?' or 'Which of the following is not always true?', or something similar.
  
  It might seem like a pedantic point, but when nearly half of the GMAT -- Data Sufficiency -- hinges on this distinction, it's a pretty important thing for a prep book to get right.

[sanju09]

Two things here:

• you should never, ever, subtract one inequality from another. You can add two inequalities which face the same way, but you can't subtract them (try subtracting the inequality 10 > 8 from the inequality 10 > 9 and you'll see that the result, 0 > 1, is complete nonsense). We aren't subtracting inequalities at all here. If we want to get information about a-b, say, we can multiply this inequality:
  
  3 < b < 12
  
  by -1. Since we're multiplying by a negative, we need to reverse the inequalities, so we find that
  
  -12 < -b < -3
  
  We can now add this to the inequality:
  
  -2 < a < 11
  
  to find that -14 < a-b < 8. If instead you want to learn about b-a, you can multiply the inequality -2 < a < 11 by -1, reversing the inequalities when you do, to get -11 < -a < 2, and then add this to the inequality 3 < b < 12. You'll find that -8 < b-a < 14.
• It's clear that C is the intended answer to this question, but if the wording of the question has been correctly transcribed in the original post, then this answer actually makes no logical sense. The question asks which inequality is "not true". Well, we know with certainty that -8 < b - a < 14; is the inequality -7 < b - a < 14 then 'not true'? We can't say; it might be true (b-a might be equal to 3, say) or it might be false (b - a might be equal to -7.5). To make logical sense, the question needs to ask 'Which of the following is not necessarily true?' or 'Which of the following is not always true?', or something similar.
  
  It might seem like a pedantic point, but when nearly half of the GMAT -- Data Sufficiency -- hinges on this distinction, it's a pretty important thing for a prep book to get right.

Absolutely

Thanks Ian