Is x negative?
(1) At least one of x and x2 is greater than x3.
(2) At least one of x2 and x3 is greater than x.
Nice One : Is x negative?
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- Ian Stewart
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If 0 < x < 1, then x^3 < x^2 < x.
If x > 1, then x < x^2 < x^3.
So when x is positive, x^2 is never the largest of the three expressions x, x^2 and x^3. Statement 1 tells us that x^3 is not the largest of the three expressions, and Statement 2 tells us x is not the largest of the three expressions. So x^2 must be the largest, and thus x cannot be positive. So the answer is C.
Note that neither statement is sufficient alone, since while x can certainly be negative using either statement, using statement 1, x could also be positive (x could be 1/2, say), and using Statement 2 x could be positive (x could be 2, say).
If x > 1, then x < x^2 < x^3.
So when x is positive, x^2 is never the largest of the three expressions x, x^2 and x^3. Statement 1 tells us that x^3 is not the largest of the three expressions, and Statement 2 tells us x is not the largest of the three expressions. So x^2 must be the largest, and thus x cannot be positive. So the answer is C.
Note that neither statement is sufficient alone, since while x can certainly be negative using either statement, using statement 1, x could also be positive (x could be 1/2, say), and using Statement 2 x could be positive (x could be 2, say).
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com
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- lunarpower
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hey, i wrote that problem!
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if you ever see a problem like this -- on which you're comparing different powers of the same variable -- then, immediately, you should think about the following number properties:
* signs
* "fractions" (between 0 and 1, or between -1 and 0) vs. "non-fractions" (greater than 1 or less than -1)
for each of these ranges of numbers -- less than -1; between -1 and 0; between 0 and 1; greater than 1 -- the powers have different properties, and so the ordering of the powers is different.
it's rather pointless (and possibly confusing) to memorize what happens to the powers. instead, just throw in a few numbers of each type and watch what they do.
if the inequalities are not strict inequalities (i.e., if they are the < or > type), then you should also mess around with the numbers 0, 1, and -1.
--
if you ever see a problem like this -- on which you're comparing different powers of the same variable -- then, immediately, you should think about the following number properties:
* signs
* "fractions" (between 0 and 1, or between -1 and 0) vs. "non-fractions" (greater than 1 or less than -1)
for each of these ranges of numbers -- less than -1; between -1 and 0; between 0 and 1; greater than 1 -- the powers have different properties, and so the ordering of the powers is different.
it's rather pointless (and possibly confusing) to memorize what happens to the powers. instead, just throw in a few numbers of each type and watch what they do.
if the inequalities are not strict inequalities (i.e., if they are the < or > type), then you should also mess around with the numbers 0, 1, and -1.
Ron has been teaching various standardized tests for 20 years.
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Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.
Yves Saint-Laurent
--
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Voit esittää kysymyksiä Ron:lle myös suomeksi
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Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.
Yves Saint-Laurent
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- ceilidh.erickson
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Not to ditto everything that Ron is saying today...
I often tell my students that whenever they see a problem with exponents and inequalities, they should draw this number line, to make sure they don't forget positives/negatives or fractions/non-fractions:
If you quickly test each of these 7 numbers, you'll have considered all of the types of numbers that behave differently with inequalities and exponents.
I often tell my students that whenever they see a problem with exponents and inequalities, they should draw this number line, to make sure they don't forget positives/negatives or fractions/non-fractions:
If you quickly test each of these 7 numbers, you'll have considered all of the types of numbers that behave differently with inequalities and exponents.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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I often tell my students that whenever they see a problem with exponents and inequalities, they should draw this number line, to make sure they don't forget positives/negatives or fractions/non-fractions:
If you quickly test each of these 7 numbers, you'll have considered all of the types of numbers that behave differently with inequalities and exponents.
Hi Ceilidh,
Can you please provide with the picture.... is the link working fine ? I am not being able to see it. Thanks in advance.
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- lunarpower
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Ceilidh's picture is at https://postimg.cc/image/6wrsd3hy1/
Ron has been teaching various standardized tests for 20 years.
--
Pueden hacerle preguntas a Ron en castellano
Potete chiedere domande a Ron in italiano
On peut poser des questions à Ron en français
Voit esittää kysymyksiä Ron:lle myös suomeksi
--
Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.
Yves Saint-Laurent
--
Learn more about ron
--
Pueden hacerle preguntas a Ron en castellano
Potete chiedere domande a Ron in italiano
On peut poser des questions à Ron en français
Voit esittää kysymyksiä Ron:lle myös suomeksi
--
Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.
Yves Saint-Laurent
--
Learn more about ron