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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Is abc at least 4? tagged by: fskilnik@GMATH ##### This topic has 2 expert replies and 0 member replies ### GMAT/MBA Expert ## Is abc at least 4? ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult GMATH practice exercise (Quant Class 14) $${\rm{Is}}\,\,abc\,\, \ge \,\,4\,\,\,?$$ $$\left( 1 \right)\,\,b + c \ge 2$$ $$\left( 2 \right)\,\,ab \ge ac \ge 4$$ Answer: ____(C)__ _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br ### GMAT/MBA Expert GMAT Instructor Joined 25 May 2010 Posted: 15348 messages Followed by: 1864 members Upvotes: 13060 GMAT Score: 790 fskilnik@GMATH wrote: GMATH practice exercise (Quant Class 14) $${\rm{Is}}\,\,abc\,\, \ge \,\,4\,\,\,?$$ $$\left( 1 \right)\,\,b + c \ge 2$$ $$\left( 2 \right)\,\,ab \ge ac \ge 4$$ Nice problem, Fabio! Statement 1: Case 1: a=4, b=1 and c=1, with the result that b+câ‰¥2 In this case, abc = 4, so the answer to the question stem is YES. Case 2: a=0, b=1 and c=1, with the result that b+câ‰¥2 In this case, abc = 0, so the answer to the question stem is NO. Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT. Statement 2: Case 1: a=4, b=1 and c=1, with the result that ab=4 and ac=4 In this case, abc = 4, so the answer to the question stem is YES. Case 2: a=-4, b=-1 and c=-1, with the result that ab=4 and ac=4 In this case, abc = -4, so the answer to the question stem is NO. Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT. Statements combined: ab â‰¥ ac â‰¥ 4 requires that a, b and c have the SAME SIGN. Since b+câ‰¥2, and b and c have the same sign, b and c must both be POSITIVE. Implication: a, b and c are ALL positive. Thus: ab â‰¥ ac (ab)/a â‰¥ (ac)/a b â‰¥ c Adding together b â‰¥ c and b+c â‰¥ 2, we get: b + b + c â‰¥ c + 2 2b â‰¥ 2 b â‰¥ 1 Inequalities constrained to positive values can be MULTIPLIED. Multiplying bâ‰¥1 and acâ‰¥4, we get: abc â‰¥ 4 Thus, the answer to the question stem is YES. SUFFICIENT. The correct answer is C. _________________ Mitch Hunt Private Tutor for the GMAT and GRE GMATGuruNY@gmail.com If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Student Review #1 Student Review #2 Student Review #3 Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now. ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 fskilnik@GMATH wrote: GMATH practice exercise (Quant Class 14) $${\rm{Is}}\,\,abc\,\, \ge \,\,4\,\,\,?$$ $$\left( 1 \right)\,\,b + c \ge 2$$ $$\left( 2 \right)\,\,ab \ge ac \ge 4$$ Hi, Mitch! Thanks for the words and for your beautiful contribution! $$abc\,\,\mathop \ge \limits^? \,\,4$$ $$\left( 1 \right)\,\,b + c\,\, \ge 2\,\,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {4,1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$ $$\left( 2 \right)\,\,ab \ge ac \ge 4\,\,\,\,\,\left\{ \matrix{ \,{\rm{(Re)Take}}\,\,\left( {a,b,c} \right) = \left( {4,1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( { - 2, - 2, - 2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$ $$\left( {1 + 2} \right)\,\,a \ne 0\,\,\,\,\,\left( {ac \ne 0} \right)\,\,\,\,::\,\,\,\,\left\{ \matrix{ \,a < 0\,\,\,\, \Rightarrow \,\,\,\,b < 0\,\,\,\,\left( {ab > 0} \right)\,\,\,\,\,and\,\,\,\,\,c < 0\,\,\,\left( {ac > 0} \right)\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\,{\rm{impossible}} \hfill \cr \,a > 0\,\,\,\, \Rightarrow \,\,\,\,b > 0\,\,\,\,\left( {ab > 0} \right)\,\,\,\,\,and\,\,\,\,\,c > 0\,\,\,\left( {ac > 0} \right) \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a,b,c\,\,\, > 0\,\,\,\,\,\left( * \right)$$ $$\left. \matrix{ ab \ge 4\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,c\,\,\left( * \right)} \,\,\,abc \ge 4c\,\,\, \hfill \cr ac \ge 4\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,b\,\,\left( * \right)} \,\,\,abc \ge 4b \hfill \cr} \right\}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\,\,2abc \ge 4\left( {b + c} \right)\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{:\,2} \,\,\,\,\,\,\,abc \ge 2\left( {b + c} \right)\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle$$ The correct answer is therefore (C). We follow the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

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