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If \(x \ne 0\) and \(\frac{x}{|x|} \lt x\), which of the following must be true?

A. \(x \gt 1\) B. \(x \gt -1\) C. \(|x| \lt 1\) D. \(|x|>1\) E. \(-1 \lt x \lt 0\)

Notice that we are asked to find which of the options MUST be true, not COULD be true.

Let's see what ranges does \(\frac{x}{|x|} \lt x\) give for \(x\). Two cases:

If \(x \lt 0\) then \(|x|=-x\), hence in this case we would have: \(\frac{x}{-x} \lt x\), which is the same as \(-1 \lt x\). But remember that we consider the range \(x \lt 0\), so \(-1 \lt x \lt 0\);

If \(x \gt 0\) then \(|x|=x\), hence in this case we would have: \(\frac{x}{x} \lt x\), which is the same as \(1 \lt x\).

So, \(\frac{x}{|x|} \lt x\) means that \(-1 \lt x \lt 0\) or \(x \gt 1\).

Only option which is ALWAYS true is B. ANY \(x\) from the range \(-1 \lt x \lt 0\) or \(x \gt 1\) will definitely be more the \(-1\).

As for other options:

A. \(x \gt 1\). Not necessarily true since \(x\) could be -0.5;

C. \(|x| \lt 1\), so \(-1 \lt x \lt 1\). Not necessarily true since \(x\) could be 2;

D. \(|x| \gt 1\), so \(x \lt -1\) or \(x \gt 1\). Not necessarily true since \(x\) could be -0.5;

E. \(-1 \lt x \lt 0\). Not necessarily true since \(x\) could be 2.

If i put X= 0.5 which is X > -1, Then the L.H.S = 1 And R.H.S = 0.5. R.H.S is not greater than L.H.S

x cannot be 0.5 because it does not satisfy given condition that \(\frac{x}{|x|} \lt x\). Please re-read the solution.

Hi Bunuel,

I am confused with the MUST be true condition, I understand that MUST be true is like SUFFICIENCY for DS problems, I do not know If I am wrong or right but for instance... If X > -1 then in a DS question the answer is YES for -1 < x < 0 and x > 1, but the answer is NO for 0 < x < 1... then it would not be sufficient... I understand that MUST is like in ALL cases the same answer (In all cases affirm the question stem), but I am confused know, could you help me?

If i put X= 0.5 which is X > -1, Then the L.H.S = 1 And R.H.S = 0.5. R.H.S is not greater than L.H.S

x cannot be 0.5 because it does not satisfy given condition that \(\frac{x}{|x|} \lt x\). Please re-read the solution.

Hi Bunuel,

I am confused with the MUST be true condition, I understand that MUST be true is like SUFFICIENCY for DS problems, I do not know If I am wrong or right but for instance... If X > -1 then in a DS question the answer is YES for -1 < x < 0 and x > 1, but the answer is NO for 0 < x < 1... then it would not be sufficient... I understand that MUST is like in ALL cases the same answer (In all cases affirm the question stem), but I am confused know, could you help me?

I think this the explanation isn't clear enough, please elaborate. This given condition would not hold true if x=1. Hence, I don't think option B is the appropriate choice. Can someone please help clarify?

valmikee: the inequality gives us either \(x >1\) or \(-1<x<0\)

Hence, in any case of satisfied solution of inequality, x will always be greater than -1 --> must be true that: \(x >-1\) (B)
_________________

[4.33] In the end, what would you gain from everlasting remembrance? Absolutely nothing. So what is left worth living for? This alone: justice in thought, goodness in action, speech that cannot deceive, and a disposition glad of whatever comes, welcoming it as necessary, as familiar, as flowing from the same source and fountain as yourself. (Marcus Aurelius)

this question is amazing. im trying to figure out - how does answer E potentially result in the value 2 if x cannot take on two distinctive values? would you mind giving me a simple example with values? thank you

this question is amazing. im trying to figure out - how does answer E potentially result in the value 2 if x cannot take on two distinctive values? would you mind giving me a simple example with values? thank you

I think Bunuel only said that (E) not necessary true since if x = 2 then the inequality is still hold.

Say x= 3 then: \(3/|3|=1 < 3,\) (correct) and so on...

Hope it's clear
_________________

[4.33] In the end, what would you gain from everlasting remembrance? Absolutely nothing. So what is left worth living for? This alone: justice in thought, goodness in action, speech that cannot deceive, and a disposition glad of whatever comes, welcoming it as necessary, as familiar, as flowing from the same source and fountain as yourself. (Marcus Aurelius)

When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|={-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|={some \ expression}\). For example: \(|5|=5\).

I think this the explanation isn't clear enough, please elaborate. I understood the algebraic method but I had a doubt. An inequality problem can be solved by plugging in values also right? If we try to plug-in values into option A, the -0.5 value which we arrive by solving algebraically, will go against the statement itself. Is it that we can't solve some inequality problems by plugging in?

I think this the explanation isn't clear enough, please elaborate. I understood the algebraic method but I had a doubt. An inequality problem can be solved by plugging in values also right? If we try to plug-in values into option A, the -0.5 value which we arrive by solving algebraically, will go against the statement itself. Is it that we can't solve some inequality problems by plugging in?

I'll try again.

The question asks if \(-1 \lt x \lt 0\) or \(x \gt 1\), then which of the following must be true.

Since \(-1 \lt x \lt 0\) or \(x \gt 1\), then it must be true to say about x that x > -1.

For example, x can be, among many other values, -0.9, -0.89292838, -0.76539, -0,5, ... (because \(-1 \lt x \lt 0\)) as well as x can be 3, \(\pi\), 4.17, \(\sqrt{71}\), ... (because \(x \gt 1\)). Any of them is greater than -1. For ANY possible x, so for ANY x from \(-1 \lt x \lt 0\) and \(x \gt 1\), it will be true to say that x is greater than -1.

Option A, which says that \(x \gt 1\), is NOT always true because if x is from \(-1 \lt x \lt 0\), say if x is -0.14, then \(x \gt 1\) will NOT be true.