If 4/x < 1/3, what is the possible range of values for x?
Answer x<0 and x>12
Also
If 4/x < -1/3, what is the possible range of values for x?
Answer -12<x<0
Please explain how these answers have been reached. Thank you in advance
inequality question ..please help
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- ganeshrkamath
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x can be either positive or negative.pcmdotcom wrote:If 4/x < 1/3, what is the possible range of values for x?
Answer [spoiler]x<0 and x>12[/spoiler]
Case 1: x is positive
4/x < 1/3
x > 12
Case 2: x is negative => x = -m _______m is a positive number
4/(-m) < 1/3
-4/m < 1/3
This is true for all positive numbers m.
So it is true for all negative numbers x.
x < 0
So [spoiler]x < 0[/spoiler] and [spoiler]x > 12[/spoiler]
Case 1: x > 0pcmdotcom wrote:Also
If 4/x < -1/3, what is the possible range of values for x?
Answer [spoiler]-12<x<0[/spoiler]
Please explain how these answers have been reached. Thank you in advance
4/x < -1/3
This will never be true.
Case 2: x < 0
Let x = -m _______m is a positive number
4/(-m) < -1/3
-4/m < -1/3
4/m > 1/3
m < 12
(-m) > -12
x > -12
So [spoiler]-12 < x < 0[/spoiler]
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- Uva@90
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Hi Pcmdotcom,pcmdotcom wrote:If 4/x < 1/3, what is the possible range of values for x?
Answer x<0 and x>12
Also
If 4/x < -1/3, what is the possible range of values for x?
Answer -12<x<0
Please explain how these answers have been reached. Thank you in advance
For the first One,
4/x < 1/3
Divide by 4 on both sides,
1/x < 1/12
Case 1: LHS can be less than RHS when x takes the negative values. So X<0
(When X takes negative LHS will be negative irrespective of value x and RHS will be positive.)
Case 2: When X takes positive value, RHS will be greater than LHS only when X>12
Reason,
Since both the sign is positive and numerator is same, Then Denominator with Higher value will be the smallest one.
Example: 1/100 < 1/10 (0.01<0.1)
If sign is negative it will be the opposite case:
Example: -(1/100) > -(1/10) (-0.01 > -0.1)
Hope it helps you.
Regards,
Uva.
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For Question 1:
4/x < 1/3
If "x" is positive .. cross multiply...
x > 12.... So overlap is x>12
if "x" is negative.. multiply both sides by "-1" .. so sign changes.
x < 12.... So overlap is x<0
4/x < 1/3
If "x" is positive .. cross multiply...
x > 12.... So overlap is x>12
if "x" is negative.. multiply both sides by "-1" .. so sign changes.
x < 12.... So overlap is x<0
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The CRITICAL POINTS are where 4/x = 1/3 and where 4/x is undefined.pcmdotcom wrote:If 4/x < 1/3, what is the possible range of values for x?
4/x = 1/3 when x=12.
4/x is undefined when x=0.
To determine the ranges where 4/x < 1/3, test one value to the left and right of each critical point.
x<0:
Plugging x=-1 into 4/x < 1/3, we get:
4/-1 < 1/3
-4 < 1/3.
This works.
Thus, x<0 is a valid range.
0<x<12:
Plugging x=1 into 4/x < 1/3, we get:
4/1 < 1/3
4 < 1/3.
Doesn't work.
Thus, 0<x<12 is not a valid range.
x>12:
Plugging x=16 into 4/x < 1/3 , we get:
4/16 < 1/3
1/4 < 1/3.
This works.
Thus, x>12 is a valid range.
Result:
4/x < 1/3 when x<0 or x>12.
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Hi pcmdotcom,
The assembled explanations properly show you the "math" behind this question.
In the big picture, this type of Quant example serves as a test of your "thoroughness", which is an aspect of your thinking that the GMAT will test repeatedly. Remember that variables aren't necessarily positive integers; they can be negative, fractions and even 0 in certain circumstances.
GMAT assassins aren't born, they're made,
Rich
The assembled explanations properly show you the "math" behind this question.
In the big picture, this type of Quant example serves as a test of your "thoroughness", which is an aspect of your thinking that the GMAT will test repeatedly. Remember that variables aren't necessarily positive integers; they can be negative, fractions and even 0 in certain circumstances.
GMAT assassins aren't born, they're made,
Rich
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4/x < 1/3
Multiply by 3x to get:
12 < x
So x > 12
Also, dividing 4 by a negative value will create a negative result (implicitly < 13)
Dividing by 0 is not possible.
Answer x<0 and x>12
If 4/x < 1/3, what is the possible range of values for x?
Answer x<0 and x>12
The x<0 is not easy to spot.
Multiply by 3x to get:
12 < x
So x > 12
Also, dividing 4 by a negative value will create a negative result (implicitly < 13)
Dividing by 0 is not possible.
Answer x<0 and x>12
If 4/x < 1/3, what is the possible range of values for x?
Answer x<0 and x>12
The x<0 is not easy to spot.