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question: x id non zero, is \(| x | < 1\) ??? now to be < 1 x must be negative, so the question become " is x negative ???

1) for a square to be < 1 the must be for example \(\frac{- 1}{2}\) ----> x is negative . suff

2) for x to be minor of a certain number that is the reciprocal i.e. : \(-2 < - 1/2\) --------> x is negative. suff

is correct or I should rethink the entire part regarding inequalities ?? because this statement should not take you more that 50 seconds to solve. otherwise you get in trouble with this exam.

Re: If x is not equal to zero, is | x | < 1 ?? [#permalink]

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30 Sep 2012, 22:50

3

This post received KUDOS

Expert's post

carcass wrote:

If x is not equal to zero, is \(| x | < 1\) ??

1) \(x^2\) \(< 1\)

2) \(| x |\) < \(\frac{1}{x}\)

I would like to know if I do in the right manner

question: x id non zero, is \(| x | < 1\) ??? now to be < 1 x must be negative, so the question become " is x negative ???

1) for a square to be < 1 the must be for example \(\frac{- 1}{2}\) ----> x is negative . suff

2) for x to be minor of a certain number that is the reciprocal i.e. : \(-2 < - 1/2\) --------> x is negative. suff

is correct or I should rethink the entire part regarding inequalities ?? because this statement should not take you more that 50 seconds to solve. otherwise you get in trouble with this exam.

Thanks

I would suggest you to keep 4 ranges in mind when dealing with squares, reciprocals etc.

You can do the question logically or by plugging in numbers.

Ques: If x is not equal to zero, is \(| x | < 1\) ?? Logically, | x | implies distance from 0 on the number line. So the question becomes "Is x a distance of less than 1 away from 0?" i.e. "Is -1 < x < 1?" given x is not 0.

Statement 1: \(x^2\) \(< 1\) Square of a number will be less than 1 only if the absolute value of the number is less than 1. This means \(| x | < 1\). The number needn't be negative. If it is positive, it should be less than 1. If it negative, it should be greater than -1. Or, do you remember how to solve inequalities using the wave? \(x^2 < 1\) is \(x^2 - 1 < 0\) which is \((x - 1)(x + 1) < 0\). This implies -1 < x < 1 but x is not 0. Or plug in numbers from the given 4 ranges. YOu will see that x must lie in -1 < x < 1. Sufficient.

Statement 2: \(| x | < \frac{1}{x}\) First of all, x cannot be negative since | x | is never negative. Since | x | is less than 1/x, 1/x must be positive. Also, x cannot be greater than 1 since then, | x | will be greater than 1/x. Or plug in numbers from the given 4 ranges. You will see that x must lie in 0 < x < 1. Sufficient.

Re: If x is not equal to zero, is | x | < 1 ?? [#permalink]

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30 Sep 2012, 06:34

2

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carcass wrote:

If x is not equal to zero, is \(| x | < 1\) ??

1) \(x^2\) \(< 1\)

2) \(| x |\) < \(\frac{1}{x}\)

I would like to know if I do in the right manner

question: x id non zero, is \(| x | < 1\) ??? now to be < 1 x must be negative, so the question become " is x negative ???

1) for a square to be < 1 the must be for example \(\frac{- 1}{2}\) ----> x is negative . suff

2) for x to be minor of a certain number that is the reciprocal i.e. : \(-2 < - 1/2\) --------> x is negative. suff

is correct or I should rethink the entire part regarding inequalities ?? because this statement should not take you more that 50 seconds to solve. otherwise you get in trouble with this exam.

Thanks

The condition \(x\) non-zero was given just because statement (2) has \(x\) in the denominator.

x id non zero, is \(| x | < 1\) ??? Not necessarily. \(x\) can be greater than \(1\) or less than \(-1\), in which case, definitely \(|x|\) is not less than \(1.\)

now to be < 1 x must be negative, so the question become " is x negative ??? NO \(x\) can be between \(0\) and \(1\).

(1) \(x^2\) \(< 1\), take square root from both sides and obtain \(|x|<1\), because \(\sqrt{x^2}=|x|\). Sufficient.

(2) Since \(|x|<\frac{1}{x}\), it follows that \(x>0\), because \(|x|>0\). \(x\) cannot be negative!!! Then, the given inequality becomes \(x<\frac{1}{x}\), or \(x^2<1\) (we can multiply both sides by \(x\), which is positive) and we are again in the situation from (1) above. Sufficient.

Answer D. _________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

(2) |x| < 1/x. Since the left hand side of the inequity (|x|) is an absolute value, which cannot be negative (actually in this case we know that its positive as we are given that \(x\neq{0}\)) then the right hand side (1/x) must also be positive, which means that \(x>0\), so \(|x|=x\).

Hence, \(|x| < \frac{1}{x}\) becomes \(x<\frac{1}{x}\). Since we know that \(x>0\) we can safely cross-multiply: \(x^2<1\). The same inequality as in (1). Sufficient.

Re: If x is not equal to zero, is | x | < 1 ?? [#permalink]

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30 Sep 2012, 07:58

Perfect Explanation, Evajager, @Carcass this is a very common mistake which I am also prone to make. Thanks @Evajager - Your explanation for {B}

Quote:

(2) Since , it follows that , because . cannot be negative!!! Then, the given inequality becomes , or (we can multiply both sides by , which is positive) and we are again in the situation from (1) above. Sufficient.

I remember skipping a Question just like this in one of my exams because I gt stuck!!! in solving - | x | < 1/x

Thanks guys you rock!!! _________________

Giving Kudos, is a great Way to Help the GC Community Kudos

Re: If x is not equal to zero, is | x | < 1 ?? [#permalink]

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30 Sep 2012, 13:29

EvaJager wrote:

carcass wrote:

If x is not equal to zero, is \(| x | < 1\) ??

1) \(x^2\) \(< 1\)

2) \(| x |\) < \(\frac{1}{x}\)

I would like to know if I do in the right manner

question: x id non zero, is \(| x | < 1\) ??? now to be < 1 x must be negative, so the question become " is x negative ???

1) for a square to be < 1 the must be for example \(\frac{- 1}{2}\) ----> x is negative . suff

2) for x to be minor of a certain number that is the reciprocal i.e. : \(-2 < - 1/2\) --------> x is negative. suff

is correct or I should rethink the entire part regarding inequalities ?? because this statement should not take you more that 50 seconds to solve. otherwise you get in trouble with this exam.

Thanks

The condition \(x\) non-zero was given just because statement (2) has \(x\) in the denominator.

x id non zero, is \(| x | < 1\) ??? Not necessarily. \(x\) can be greater than \(1\) or less than \(-1\), in which case, definitely \(|x|\) is not less than \(1.\)

now to be < 1 x must be negative, so the question become " is x negative ??? NO \(x\) can be between \(0\) and \(1\).

(1) \(x^2\) \(< 1\), take square root from both sides and obtain \(|x|<1\), because \(\sqrt{x^2}=|x|\). Sufficient.

(2) Since \(|x|<\frac{1}{x}\), it follows that \(x>0\), because \(|x|>0\). \(x\) cannot be negative!!! Then, the given inequality becomes \(x<\frac{1}{x}\), or \(x^2<1\) (we can multiply both sides by \(x\), which is positive) and we are again in the situation from (1) above. Sufficient.

Answer D.

Correction: in (2) and we are again in the situation from (1) above. - not exactly, but similar as we have \(x^2<1\) but in addition \(x>0.\) In (1) \(x\) could also be negative. The conclusion is still correct, as now \(|x|=x<1.\) _________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

Re: If x is not equal to zero, is | x | < 1 ?? [#permalink]

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01 Oct 2012, 04:24

Expert's post

VeritasPrepKarishma wrote:

carcass wrote:

If x is not equal to zero, is \(| x | < 1\) ??

1) \(x^2\) \(< 1\)

2) \(| x |\) < \(\frac{1}{x}\)

I would like to know if I do in the right manner

question: x id non zero, is \(| x | < 1\) ??? now to be < 1 x must be negative, so the question become " is x negative ???

1) for a square to be < 1 the must be for example \(\frac{- 1}{2}\) ----> x is negative . suff

2) for x to be minor of a certain number that is the reciprocal i.e. : \(-2 < - 1/2\) --------> x is negative. suff

is correct or I should rethink the entire part regarding inequalities ?? because this statement should not take you more that 50 seconds to solve. otherwise you get in trouble with this exam.

Thanks

I would suggest you to keep 4 ranges in mind when dealing with squares, reciprocals etc.

You can do the question logically or by plugging in numbers.

Ques: If x is not equal to zero, is \(| x | < 1\) ?? Logically, | x | implies distance from 0 on the number line. So the question becomes "Is x a distance of less than 1 away from 0?" i.e. "Is -1 < x < 1?" given x is not 0.

Statement 1: \(x^2\) \(< 1\) Square of a number will be less than 1 only if the absolute value of the number is less than 1. This means \(| x | < 1\). The number needn't be negative. If it is positive, it should be less than 1. If it negative, it should be greater than -1. Or, do you remember how to solve inequalities using the wave? \(x^2 < 1\) is \(x^2 - 1 < 0\) which is \((x - 1)(x + 1) < 0\). This implies -1 < x < 1 but x is not 0. Or plug in numbers from the given 4 ranges. YOu will see that x must lie in -1 < x < 1. Sufficient.

Statement 2: \(| x | < \frac{1}{x}\) First of all, x cannot be negative since | x | is never negative. Since | x | is less than 1/x, 1/x must be positive. Also, x cannot be greater than 1 since then, | x | will be greater than 1/x. Or plug in numbers from the given 4 ranges. You will see that x must lie in 0 < x < 1. Sufficient.

Answer (D)

This is the same reasoning that I followed in the first instance with some errors but the path was correct. Specifically the stimulus evaluation : \(| x | < 1\) ------ in this scenario the first one is \(x < 1\) AND \(-x < 1\) so \(x > -1\) ------> \(- 1 < x < 1\)

Re: If x is not equal to zero, is |x| < 1 ? [#permalink]

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Re: If x is not equal to zero, is |x| < 1 ? [#permalink]

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