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M28-04

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Math Expert
Joined: 02 Sep 2009
Posts: 47967

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16 Sep 2014, 01:28
00:00

Difficulty:

35% (medium)

Question Stats:

64% (00:28) correct 36% (00:47) wrong based on 44 sessions

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What is the sum of integers $$a$$ and $$b$$ ?

(1) $$|a| = -|b|$$

(2) $$|b| = -|a|$$

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Math Expert
Joined: 02 Sep 2009
Posts: 47967

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16 Sep 2014, 01:28
Official Solution:

Both statements are saying the exact same thing: $$|a|+|b|=0$$, this to be true, both $$a$$ and $$b$$ must equal to zero.

Why is that? Absolute value is always non-negative - $$|\text{some expression}| \geq 0$$, which means that absolute value is either zero or positive. We have that the sum of two absolute values, or the sum of two non-negative values equals to zero: $$\text{non-negative} + \text{non-negative} = 0$$, obviously both must be zero this equation to hold true.

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Joined: 01 Nov 2016
Posts: 68
Concentration: Technology, Operations

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18 Apr 2017, 18:20
Bunuel wrote:
Official Solution:

Both statements are saying the exact same thing: $$|a|+|b|=0$$, this to be true, both $$a$$ and $$b$$ must equal to zero.

Why is that? Absolute value is always non-negative - $$|\text{some expression}| \geq 0$$, which means that absolute value is either zero or positive. We have that the sum of two absolute values, or the sum of two non-negative values equals to zero: $$\text{non-negative} + \text{non-negative} = 0$$, obviously both must be zero this equation to hold true.

In other words, is this the logic? Re-arrange the statements to be:

(1) |a| + |b| = 0

(2) |b| + |a| = 0

If you add two positive values and they equal zero, then both values have to be zero
Math Expert
Joined: 02 Sep 2009
Posts: 47967

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18 Apr 2017, 21:22
joondez wrote:
Bunuel wrote:
Official Solution:

Both statements are saying the exact same thing: $$|a|+|b|=0$$, this to be true, both $$a$$ and $$b$$ must equal to zero.

Why is that? Absolute value is always non-negative - $$|\text{some expression}| \geq 0$$, which means that absolute value is either zero or positive. We have that the sum of two absolute values, or the sum of two non-negative values equals to zero: $$\text{non-negative} + \text{non-negative} = 0$$, obviously both must be zero this equation to hold true.

In other words, is this the logic? Re-arrange the statements to be:

(1) |a| + |b| = 0

(2) |b| + |a| = 0

If you add two positive values and they equal zero, then both values have to be zero

Two non-negative values. The sum of positive numbers cannot be 0.
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Joined: 04 Feb 2014
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10 Jul 2018, 22:19
Bunuel
I understand the arrangement of the equation made above but I was considering the following:
(1) if a>0, a=-b; -b+b=0
BUT if a<0, -a=-b; a=b; and then the sum is a+a=2a (or b+b=2b)
in this case the answer would be not sufficient. Where is the mistake here?
Thank you
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Joined: 02 Sep 2009
Posts: 47967

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11 Jul 2018, 00:05
1
NatalieStarr wrote:
Bunuel
I understand the arrangement of the equation made above but I was considering the following:
(1) if a>0, a=-b; -b+b=0
BUT if a<0, -a=-b; a=b; and then the sum is a+a=2a (or b+b=2b)
in this case the answer would be not sufficient. Where is the mistake here?
Thank you

If a > 0 and b > 0, then |a| = -|b| transforms to a = -b --> a + b = 0. This case is not possible: the sum of two positive values cannot be 0.
If a > 0 and b < 0, then |a| = -|b| transforms to a = -(-b) --> a = b. This case is not possible: positive a (a > 0) cannot equal to negative b (b < 0).
If a < 0 and b < 0, then |a| = -|b| transforms to -a = -(-b) --> a + b = 0. This case is not possible: the sum of two negative values cannot be 0.
If a < 0 and b > 0, then |a| = -|b| transforms to -a = -b --> a = b. This case is not possible: negative a (a < 0) cannot equal to positive b (b > 0).

The only case left is a = b = 0.
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Re: M28-04 &nbs [#permalink] 11 Jul 2018, 00:05
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