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# If |m|/m < m which of the following must be true about m?

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If |m|/m < m which of the following must be true about m?  [#permalink]

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28 Mar 2020, 18:19
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GMATBusters’ Quant Quiz Question -4

If $$\frac{|m|}{m}<m$$ which of the following must be true about $$m$$?

(A) $$m>1$$

(B) $$m>-2$$

(C) $$|m|<1$$

(D) $$|m|=1$$

(E) $$|m|^2>1$$

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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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28 Mar 2020, 18:41
If $$\frac{|m|}{m}<m$$ which of the following must be true about mm?

(A) m>1

(B) m>−2

(C) |m|<1

(D) |m|=1

(E) $$|m|^2>1$$

$$\frac{|m|}{m}<m$$

$$\frac{|m|}{m}-m<0$$
$$\frac{(|m|-m^2)}{m}<0$$

Case i) m<0; |m|-m^2>0
$$|m| > m^2$$
Squaring both sides
$$m^2 > m^4$$
$$m^4 - m^2 < 0$$
$$m^2(m+1)(m-1) < 0$$
-1<m<1 when m < 0 =>
-1<m<0 ---------------- i)

Case ii) m>0; |m|-m^2<0
$$|m| < m^2$$
Squaring both sides
$$m^2 < m^4$$
$$m^4 - m^2 > 0$$
$$m^2(m+1)(m-1) > 0$$
m > 1 or m < -1 when m > 0 =>
m > 1 ---------------- ii)

Only option satisfying both the conditions of i) and ii) is option B

i.e. m > -2

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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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28 Mar 2020, 20:49
If |m|/m<m, which of the following must be true about m?

If m = 0, 0/0 <0, not possible
If m = 0.5, then 0.5/0.5<0.5, 1<0.5, not possible
If m = Positive = 1, then 1/1<1, 1<1, so m can not be 1.
If m = Positive = 2, then 2/2<2, 1<2, so m can be 2 or greater than 2.
If m= negative = -1, then 1/-1<-1, -1<-1, not possible
If m = -0.5, then 0.5/-0.5<-0.5, -1<-0.5. So m can be between 0 to -1.
If m= negative = -2, then 2/-2<-2, -1<-2, again not possible
Last one m=-3, then 3/-3<-3, -1<-3, not possible.

So, m can be between -1 and 0 or > 1, as per option only B, which states that m can be anything above -2, fits the condition.

Ans. B.
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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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28 Mar 2020, 21:07
If |m|/m<m, which of the following must be true about m?
|m|/m - m <0
(|m|-m^2)/m <0
If |m|>m^2; m<0; |m| = -m; -m>m^2; m^2+m<0 ; m(m+1)<0; -1<m<0; |m|<1
But if |m|<m^2; m>0; |m| = m; m<m^2; m^2-m>0 ; m(m-1)>0; m>1;
m = (-1,0)U(1,infinity)
-1<m<0 or m>1 are solutions

(A) m>1
-1<m<0 or m>1
Since -1<m<0 is possible
COULD BE TRUE but MUST NOT BE TRUE

(B) m>−2
-1<m<0 or m>1
m MUST BE >-2
MUST BE TRUE

(C) |m|<1
-1<m<0 or m>1
If -1<m<0; |m|<1
But if m>1; |m|>1
COULD BE TRUE but MUST NOT BE TRUE

(D) |m|=1
-1<m<0 or m>1
|m| = 1 is NOT Possible
MUST NOT BE TRUE

(E) |m|^2>1
|m|>1
-1<m<0 or m>1
If -1<m<0; |m|^2<1
But if m>1; |m|^2>1
COULD BE TRUE but MUST NOT BE TRUE

IMO B
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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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28 Mar 2020, 22:06
Ans A
If |m|/m<m
then m would be greater than 1
let m=2, then 2/2=1<m

or m would be between 0 and -1
let m=-1/2, then , 1/2/-1/2=-1<m

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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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28 Mar 2020, 22:42
If |m|m<m|m|m<m which of the following must be true about mm?

(A) m>1m>1

(B) m>−2m>−2

(C) |m|<1|m|<1

(D) |m|=1|m|=1

(E) |m|2>1

Let's test some numbers, m, and plug them into f(m) = abs(m)/m. The key domains to test will be negative numbers, and fractions, and combinations.

m=-2, f(m) = 2/-2=-1, f(m)>m
m=-1/2, f(m) = -1, f(m)< m
m=1/2, f(m) = 1, f(m) > m
m = 2, f(m) = 1, f(m) < m

therefore, we need an answer that accounts for m being negative fractions or greater than 1. m > -2, which is (b)
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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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29 Mar 2020, 04:37
as only x>-2 will have all the values of the equation
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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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29 Mar 2020, 04:52
We can see with a value -$$\frac{1}{2}$$,we satisfy the given equation
This value suffices (C) |m|<1
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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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29 Mar 2020, 05:10
|m|/m is defined so m≠0

If m is positive, |m|=m and so |m|/m<m means m>1

If m is negative, |m|=-m and so |m|/m<m means m>-1

We see that in both cases, m>-2

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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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29 Mar 2020, 05:17
For must be true questions, the quickest approach is the prove the answer choices wrong.

B) m>-2 does not hold for value m= -1
C) |m|<1 does not hold for value m= 1/2
D) |m|=1 does not hold for value m= -1
E) |m|^2 >1 does not hold for value m= -2

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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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29 Mar 2020, 05:48
Option (B)
Let's break the inequality |m|/m<m into two cases
Case 1: m>0
In such a case, |m|=m
Therefore the LHS of the equation reduces to |m|/m=m/m=1
Putting the LHS back, this leads to
1<m
Case 2: m<0
In such a case, |m|=-m
Therefore the LHS of the equation reduces to -m/m= -1
Putting this back, this leads to -1<m. Implying that solution in this case is -1<m<0 (because this case if for m<0)

Combining the two cases, we get the solution space to be
(-1,0)U(1,inf).
The only option that satisfies this is option (B) m>-2

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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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29 Mar 2020, 05:54
|m|/m<m

If m>0, |m| = m => 1<m
If m<0, |m|= -m => -1<m => -1<m<0

Only option B is true for both above equation. Hence B is the answer
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Re: If |m|/m < m which of the following must be true about m?  [#permalink]

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29 Mar 2020, 14:26
|m| < m^2
When m > 1, like m = 2, this is true.
Again when, m < -1, this holds true. Only for values of m between - 1 to 1 this doesn't work.
So, |m|^2 > 1.
Re: If |m|/m < m which of the following must be true about m?   [#permalink] 29 Mar 2020, 14:26