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GMAT Algebra http://www.800score.com/forum/viewtopic.php?f=3&t=3414 
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Author:  questioner [ Sat Apr 07, 2012 6:47 am ] 
Post subject:  GMAT Algebra 
For any a and b that satisfy a – b = b – a and a > 0, then a + 3 + b + b – a + ab = A. ab + 3 B. ab + 3 C. ab + 2b + 3 D. ab + 2b – 2a – 3 E. ab + 2b + 3 (E) We know that a – b = b – a, so b – a > 0, or b > a. We also know that a > 0, so b > 0. Knowing that both a and b are positive we can easily simplify: a + 3 + b + b – a + ab = a + 3 + b + b – a + ab = ab + 2b + 3 The right answer is choice (E).  How would the equation be exactly transformed given that either a or b or both are negative? Thank you in advance for your help. 
Author:  Gennadiy [ Sat Apr 07, 2012 7:07 am ] 
Post subject:  Re: GMAT Algebra 
Quote: How would the equation be exactly transformed given that either a or b or both are negative? I'd like to stress that this is a hypothetical situation, because in the question we show that b > a > 0.So let's try the proposed conditions. a + 3 + b + b – a + ab = ? 1) b < 0 In this case we can simplify b = b, because b > 0. But we can NOT simplify any other absolute value, because we know nothing about a. 2) a < 0 In this case we can NOT simplify any absolute value, because a + 3 can be a + 3 (if a is from 3 to 0) OR a + 3 can be a – 3 (if a is less than 3) Any other absolute value contains b and we know nothing about it. 3) a < 0 and b < 0 In this case we can simplify: b = b, because b > 0; ab = ab, because ab > 0. We can NOT simplify: a + 3 for the same reasons as in 2) b – a can be b – a (if b > a) b – a can be b + a (if b < a). You may try to plug some numbers in the absolute values we could NOT simplify: 3) a < 0 and b < 0 a = 1 < 0 , b = 4 < 0 a + 3 = 1 + 3 = 1 + 3 = 2 in this case we used a + 3 = a + 3, because (a = 1 is between 3 and 0) b – a = 4 – (1) = (4) + (1) = 3 in this case we used b – a = b + a, because (b = 4 < 1 = a) a = 4 < 0 , b = 1 < 0 a + 3 = 4 + 3 = (4) – 3 = 1 in this case we used a + 3 = a – 3, because (a = 1 is less than 3) b – a = 1 – (4) = 1 – (4) = 3 in this case we used b – a = b – a, because (b = 1 > 4 = a) 
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