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If a b and ab = ba, which of the following statements [#permalink]
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26 Mar 2013, 09:07
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If a ≠ b and ab = ba, 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.
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Re: If a ≠ b and ab = ba, which of the following statements [#permalink]
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26 Mar 2013, 09:32
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I think the best way here is using real numbers I) a < 0 a=1, b=5 15=51 4=4, so I is not always true II) a + b < 0 a=3,b=4 34=43 1=1, so II is not always true III) a < b ab = ba if \(a>b\) then \(ab = ab\) and doesn't equal ba if \(b>a\) then \(ab = (ab) = a+b = ba\) so III is true
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Re: If a ≠ b and ab = ba, which of the following statements [#permalink]
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27 Mar 2013, 00:26
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The given conditions are : i) a is not equal to b ,i,e ab is non zero. ii) a b  = ba ,i,e (ab). So ,considering the above conditions, a  b < 0 => a < b.
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Re: If a ≠ b and ab = ba, which of the following statements [#permalink]
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27 Mar 2013, 05:06
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If a ≠ b and ab = ba, 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 \(ab = ba=(ab)\), then \(ab\leq{0}\) > \(a\leq{b}\). 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. Answer: C. For more check here: mathabsolutevaluemodulus86462.html
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Re: If a b and ab = ba, which of the following statements [#permalink]
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15 Jun 2013, 11:24
Because ab = ba, could we say that ba 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:
ab = ba so:
ab = ba 2a = 2b a=b (which isn't true as the stem tells us it isn't)
OR
a+b=ba 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 ab = ba, which of the following statements [#permalink]
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15 Jun 2013, 11:27
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WholeLottaLove wrote: Because ab = ba, could we say that ba 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:
ab = ba so:
ab = ba 2a = 2b a=b (which isn't true as the stem tells us it isn't)
OR
a+b=ba 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 ab = ba, which of the following statements [#permalink]
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15 Jun 2013, 17:58
So, in other words, I.) ab = ba II.) ba is positive because it is equal to an absolute value III.) b must be greater than a because ba is positive IV.) ab must be negative V.) ab = (ab) VI.) ab ≤ 0 VII.) a ≤ b Zarrolou wrote: WholeLottaLove wrote: Because ab = ba, could we say that ba 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:
ab = ba so:
ab = ba 2a = 2b a=b (which isn't true as the stem tells us it isn't)
OR
a+b=ba 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 ab = ba, which of the following statements [#permalink]
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15 Jun 2013, 23:32
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WholeLottaLove wrote: So, in other words,
I.) ab = ba II.) ba is positive because it is equal to an absolute value III.) b must be greater than a because ba is positive IV.) ab must be negative V.) ab = (ab) VI.) ab ≤ 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 ab = ba, which of the following statements [#permalink]
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16 Jun 2013, 07:58
Ahh  I forgot about that. Thanks! Zarrolou wrote: WholeLottaLove wrote: So, in other words,
I.) ab = ba II.) ba is positive because it is equal to an absolute value III.) b must be greater than a because ba is positive IV.) ab must be negative V.) ab = (ab) VI.) ab ≤ 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 ab = ba, which of the following statements [#permalink]
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02 Sep 2014, 16:42
Another thing to remember is that, because of the absolute value
ab ≤ ba
because the absolute value either turns a negative positive or leaves a positive the same..



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If a b and ab = ba, which of the following statements [#permalink]
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25 Jul 2016, 01:57
matspring wrote: If a ≠ b and ab = ba, 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 :> abanything that comes out of a mod is positive ; therefore ab=postive ..........equation 1Now lets check the RHS > ba The question stem tells us that ab=ba .............equation 2Rearranging equation 2 gives us ba =ab .............. equation 3Equation 1 tells that ab is positive Put value of ab from equation 1 into equation 3 ba = ab=positive ba =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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