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if x<1 and x ≠ 0

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if x<1 and x ≠ 0 [#permalink] New post 20 Jan 2016, 18:21
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73% (00:32) correct 26% (00:23) wrong based on 121 sessions
x<1 and x ≠ 0

Quantity A
Quantity B
\(x^2 + 1\)
\(x^3 + 1\)


A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.




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Question: 2
Page: 457
Difficulty: medium
[Reveal] Spoiler: OA

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Re: if x<1 and x ≠ 0 [#permalink] New post 21 Jan 2016, 08:39
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Solution

A conceptual approach is the better way here to attack the question

\(x^2\) no matter what is always positive

on the other hand \(x^3\) no. could be positive or negative: it depends on the sign of the number

We do know that\(x<1\) so X could be positive or negative as well. But the \(A\) quantity is always positive and \(>\) than quantity \(B\) that could be positive or negative

The answer is \(A\)
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Re: if x<1 and x ≠ 0 [#permalink] New post 09 Jun 2016, 09:26
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sandy wrote:
x<1 and x ≠ 0

Quantity A
Quantity B
\(x^2 + 1\)
\(x^3 + 1\)


A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.



Given:
Quantity A: x² + 1
Quantity B: x³ + 1

Subtract 1 from both quantities ...
Quantity A: x²
Quantity B: x³

NOTE: Since x ≠ 0, we know that x² MUST be POSITIVE.
This allows us to divide both quantities by x² to get....
Quantity A: 1
Quantity B: x

Since it is given that x < 1, we can be certain that quantity A is greater.

Answer: A

For more on this technique see...

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Re: if x<1 and x ≠ 0 [#permalink] New post 30 Jun 2018, 09:48
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This problem was a good candidate for a picking numbers strategy. Because x<1 and x not equal to zero, I picked the numbers 1/2, -1/2 and -1. In each of those cases, A was the largest quantity, so I was pretty confident in choosing A as the correct answer.
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Re: if x<1 and x ≠ 0 [#permalink] New post 30 Jun 2018, 15:35
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Roadrunner wrote:
This problem was a good candidate for a picking numbers strategy. Because x<1 and x not equal to zero, I picked the numbers 1/2, -1/2 and -1. In each of those cases, A was the largest quantity, so I was pretty confident in choosing A as the correct answer.


Nice! Correct strategy.

Usually we might even pick a an integer other than +/- 1 in such cases. Such as -2 in this case.

The idea is to validate if the hypothesis holdds for both integers and fractions alike.
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Re: if x<1 and x ≠ 0 [#permalink] New post 09 Jul 2018, 18:40
sandy wrote:
x<1 and x ≠ 0

Quantity A
Quantity B
\(x^2 + 1\)
\(x^3 + 1\)


A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.




Practice Questions
Question: 2
Page: 457
My Answer is A
Difficulty: medium
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Re: if x<1 and x ≠ 0 [#permalink] New post 05 Aug 2018, 16:18
I did substitution for this. But we have to be careful. I picked -1, -0.5 and +0.5. We have to make sure we cover all bases. In all cases, here, A was larger. It is also important to remember that during exam, because of time crunch and panic we might miss certain numbers.
Re: if x<1 and x ≠ 0   [#permalink] 05 Aug 2018, 16:18
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