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# If a b and |a-b| = b-a, which of the following statements

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Re: If a ≠ b and |a-b| = b-a, which of the following statements [#permalink]
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The given conditions are :
i) a is not equal to b ,i,e a-b is non zero.
ii) |a -b | = b-a ,i,e -(a-b).
So ,considering the above conditions,
a - b < 0 => a < b.
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
Because |a-b| = b-a, could we say that b-a is positive (because it is equal to an abs. val.) and therefore, b must be greater than a?

Also, I first tired to solve this problems by:

|a-b| = b-a so:

a-b = b-a
2a = 2b
a=b
(which isn't true as the stem tells us it isn't)

OR

-a+b=b-a
0=0

But I'm not sure how to interpret that result. Is that a valid way to solve the problem?
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
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WholeLottaLove wrote:
Because |a-b| = b-a, could we say that b-a is positive (because it is equal to an abs. val.) and therefore, b must be greater than a?

Also, I first tired to solve this problems by:

|a-b| = b-a so:

a-b = b-a
2a = 2b
a=b
(which isn't true as the stem tells us it isn't)

OR

-a+b=b-a
0=0

But I'm not sure how to interpret that result. Is that a valid way to solve the problem?

The second result tells you that whatever value $$a$$ and $$b$$ have, that equation will always be true: $$0=0$$ always.

0=0 means that that case will always hold, hence that case (b>a) will always be "true"
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
So, in other words,

I.) |a-b| = b-a
II.) b-a is positive because it is equal to an absolute value
III.) b must be greater than a because b-a is positive
IV.) a-b must be negative
V.) |a-b| = -(a-b)
VI.) a-b ≤ 0
VII.) a ≤ b

Zarrolou wrote:
WholeLottaLove wrote:
Because |a-b| = b-a, could we say that b-a is positive (because it is equal to an abs. val.) and therefore, b must be greater than a?

Also, I first tired to solve this problems by:

|a-b| = b-a so:

a-b = b-a
2a = 2b
a=b
(which isn't true as the stem tells us it isn't)

OR

-a+b=b-a
0=0

But I'm not sure how to interpret that result. Is that a valid way to solve the problem?

The second result tells you that whatever value $$a$$ and $$b$$ have, that equation will always be true: $$0=0$$ always.

0=0 means that that case will always hold, hence that case (b>a) will always be "true"
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
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WholeLottaLove wrote:
So, in other words,

I.) |a-b| = b-a
II.) b-a is positive because it is equal to an absolute value
III.) b must be greater than a because b-a is positive
IV.) a-b must be negative
V.) |a-b| = -(a-b)
VI.) a-b ≤ 0
VII.) a ≤ b

Yes, perfect. Just remember that we are told that $$a\neq{b}$$ so

VII)$$a<b$$
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
Ahh - I forgot about that. Thanks!

Zarrolou wrote:
WholeLottaLove wrote:
So, in other words,

I.) |a-b| = b-a
II.) b-a is positive because it is equal to an absolute value
III.) b must be greater than a because b-a is positive
IV.) a-b must be negative
V.) |a-b| = -(a-b)
VI.) a-b ≤ 0
VII.) a ≤ b

Yes, perfect. Just remember that we are told that $$a\neq{b}$$ so

VII)$$a<b$$
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
Another thing to remember is that, because of the absolute value

|a-b| ≤ b-a

because the absolute value either turns a negative positive or leaves a positive the same..
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If a b and |a-b| = b-a, which of the following statements [#permalink]
matspring wrote:
If a ≠ b and |a-b| = b-a, which of the following statements must be true ?

I. a < 0
II. a + b < 0
III. a < b

(A) None
(B) I only
(C) III only
(D) I and II
(E) II and III

Thank's in advance for helping to solve the problem, the OA should be ( C ) , but I'm not sure 100% about it; a friend gave to me several GMAT exercises for training.

Lets interpret the LHS first :----> |a-b|
anything that comes out of a mod is positive ; therefore |a-b|=postive ..........equation 1
Now lets check the RHS -------> b-a
The question stem tells us that |a-b|=b-a .............equation 2
Rearranging equation 2 gives us b-a =|a-b| .............. equation 3

Equation 1 tells that |a-b| is positive
Put value of |a-b| from equation 1 into equation 3
b-a = |a-b|=positive
b-a =positive

This is only possible when b > a
we cannot say anything with mathematical surety except that b>a

Hence the answer is C (III only)
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
Since, it is given that
|a-b| = b-a, we can infer following from Absolute Value -
|a-b| = -(a-b) = b-a => a-b must be negative or 0
i.e. a-b < 0 (a=B, so a-b cannot be 0)
=> a<b Hence, only option III satisfies.
Now, let us see other options -

I. a < 0 even without plugging numbers, we can see this cannot be true always. No info ob B, and depending upon b the equation may vary.
II. a + b < 0 B=Not neccessarily tru, since we already know that only a<b is sufficient. It foes not matter whether a is +ve or _ve as long as it is less than b.
III. a < b

Hence, C only option III works.
Tip: Sometimes, I do not tend to plug numbers because they eat away time, and questions like (oops such as ) these can be solve without this strategy.
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
Bunuel wrote:
If a ≠ b and |a-b| = b-a, which of the following statements must be true ?

I. a < 0
II. a + b < 0
III. a < b

(A) None
(B) I only
(C) III only
(D) I and II
(E) II and III

Absolute value properties:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$;

Thus, according to the above, since $$|a-b| = b-a=-(a-b)$$, then $$a-b\leq{0}$$ --> $$a\leq{b}$$. Since we also know that $$a\neq{b}$$, then we have that $$a<b$$. So, III is always true.

As for the other options:
I. a < 0 --> not necessarily true, consider a=1 and b=2.
II. a + b < 0 --> not necessarily true, consider a=-2 and b=-1.

For more check here: https://gmatclub.com/forum/math-absolute ... 86462.html

I am really struggling with the below line

Thus, according to the above, since $$|a-b| = b-a=-(a-b)$$, then $$a-b\leq{0}$$ --> $$a\leq{b}$$. Since we also know that $$a\neq{b}$$, then we have that $$a<b$$

Why does $$|a-b| = b-a=-(a-b)$$?

I thought you have to check if | a-b| > 0 and < 0?
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
|a-b| = -(a-b) <---- given

Also, we know that if |x| = -x then x<0 therefore, (a-b) < 0 -----> a<b
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
|a-b| = |b-a| (GMAT property)

Let x=|b-a|

--> |x|=x
As per the graph of the two, only possible when x>=0

--> |b-a| >= 0 (we can ignore the = as b not equals a)
--> b-a>0 ---> b>a

Hence the answer is C (Only III follows)
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
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Re: If a b and |a-b| = b-a, which of the following statements [#permalink]
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