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If x is a real number and (x+5)(x-3) is a negative real number, then the value of which of the following expressions must also be a negative number?
I. 3-x
II. 3x - 7
III. (x+7)(x-3)
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
2023-12-01_19-43-03.png [ 38.53 KiB | Viewed 22878 times ]
I. 3-x
II. 3x - 7
III. (x+7)(x-3)
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
Attachment:
2023-12-01_19-43-03.png [ 38.53 KiB | Viewed 22878 times ]
17 Nov 2023, 09:32
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ArjunKumar25
If x is a real number and (x+5)(x-3) is a negative real number, then the value of which of the following expressions must also be a negative number?
I. 3-x
II. 3x - 7
III. (x+7)(x-3)
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
Given (x+5)(x-3) < 0.
The "roots", in ascending order, are -5 and 3, which gives us 3 ranges:
- x < -5;
-5 < x < 3;
x > 3.
Next, test an extreme value for x: if x is some large enough number, say 10, then both multiples will be positive, giving a positive result for the whole expression. So when x > 3, the expression is positive. Now the trick: as in the 3rd range, the expression is positive, then in the 2nd it'll be negative, in the 1st it'll be positive: (+ - +). So, the ranges when the expression is negative are: -5 < x < 3.
Taking in account this range for x lets analyze each option.
I. 3 - x. This expression will always be positive for -5 < x < 3. Discard.
II. 3x - 7. This expression can be both negative and positive for -5 < x < 3. For example, if x = 0, it will be negative. However, is x = 2.9, it will be positive. Discard.
III. (x + 7)(x - 3). For -5 < x < 3, the first multiple, x + 7, will always be positive, while the second multiple, x - 3, will always be negative, hence their product will be negative.
Answer: C.
25 Jun 2024, 01:17
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hugogva
The question is based on the wavy line method even if it doesn't occur to you right away.
Wavy line method discussed here: https://youtu.be/PWsUOe77__E
If you understand why the method is what it is, then this question can be solved within 30 secs
Given: (x+5)(x-3) < 0
So -5 < x < 3
I. 3-x
Certainly positive for negative values of x. Eliminate.
II. 3x - 7
For 3x - 7 < 0, we must have x < 7/3. But x can lie between 7/3 and 3 too hence 3x-7 may not always be negative.
GMAT has graciously mentioned here that x is a real number ensuring that you don't consider it to be an integer only.
III. (x+7)(x-3)
For (x+7)(x-3) < 0
Again using wavy line, for this to be negative, x must lie between -7 < x < 3.
When we know that x lies between -5 and 3, we know that it automatically lies within -7 and 3.
Hence this will always be negative.
Answer (C)
