Inequality / min max strategy question

Question

Can someone walk me through the below? Thanks so much.

4 < (7-x)/3

Quantity A
Maximum value of -(5-x)

Quantity B
Maximum value of 2x

AndrewN · Answer

esonrev  - This looks way more like a Quantitative Comparison on the GRE® than it does a GMATTM-style question. You might want to try posting it on GREPrepClub instead. However, in the interest of helping out, here are my two cents:

1) Work the given inequality algebraically.
 4 < \frac{(7 - x)}{3} 
 12 < (7 - x)  (multiply both sides by 3 to get rid of the fraction)
 5 < - x  (subtract 7 from both sides to isolate the unknown)
 -5 > x  (divide by -1 and flip the inequality, or, alternatively, add  x  to both sides and subtract 5)

Now, if -5 is  greater than   x , then  x  can be -5.01, -18, or whatever, as long as it does not get closer to 0 or anything positive than -5 does.  Quantity A  can therefore be reinterpreted as saying, "Maximum value of -(5 -  )." This  must be  a net negative value. If we were to insert the forbidden number, -5, we would get -(5 -(-5)), or -(5 + 5), or -(10), or -10. Thus, it is conceivable that if we were to test a number ever so slightly smaller than -5, say, -5.000000000000000000001, we could derive an answer that would be ever so slightly below the pivotal value of -10.

Meanwhile,  Quantity B  could be reinterpreted as, "Maximum value of 2( )." This  must also be  a net negative value, since a positive 2 times a negative number will yield a negative product. Once again, if we test our forbidden -5, we would get 2(-5), or -10. We could drag out our -5 with a decimal, twenty zeros, and a 1 at the end again, but we can see quite plainly that our maximum value will be ever so slightly below the pivotal value of -10.

What to do now? The answer would appear to be (C). However, with a little number sense, we can disprove such a notion. What if we were to try -5.000001 for  x  instead? In such a case, we would get -10.000001 for Quantity A. But look at what happens in Quantity B. 2(-5.000001) = -10.000002. This number is so close to the first one that it could split a hair, but that still means that the former number is larger. In fact, regardless of how many zeros we might want to place between our decimal and a 1 at the end, by doubling the value in the second expression, we effectively nudge it further from 0 than we would by taking the unadulterated number from the first expression. Taken together, then, we can conclude that  (A), Quantity A will be greater . With no stipulation on what  x  may be, in terms of whether it needs to be an integer or just some number, it can help to test the restricted value for a quick comparison. (That is, a -5 with twenty zeros trailing a decimal with a 1 on the end will behave in many ways just like a -5, so we can use that integer value for a snapshot of how the expressions will behave right before the cutoff value.)

I hope that helps. Please let me know if you would like to discuss the matter further. Good luck with your studies, whether you are preparing for the GRE® or GMATTM.

- Andrew

Inequality / min max strategy question

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