If 4/x < 1/3 , what is the possible range of values of x?

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If 4/x < 1/3 , what is the possible range of values of x? [#permalink]

25 Aug 2012, 01:47

If \frac{4}{x} <\frac{1}{3} , what is the possible range of values of x?

The question is from an MGMAT guide and the solution provided multiplies the inequality by x and flips the sign to arrive at the solution x < 0 or x > 12 . However the way I naturally would solve the given equation is:
\frac{4}{x} - \frac{1}{3} < 0
\frac{(12 - x)}{(3x)} < 0

Two cases:
a) 12 - x < 0 &3x > 0 :: x > 12 & x > 0 therefore x > 12
b) 12 - x > 0 & 3x < 0 :: x < 12 & x < 0 therefore x < 12
Hence my solution: x < 12 or x > 12

Can someone kindly explain where I went wrong here?
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Re: If 4/x < 1/3 , what is the possible range of values of x? [#permalink]

25 Aug 2012, 02:46

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harshvinayak wrote:

When we consider the case when 12 - x > 0 and 3x < 0, we have: x<12and x<0, therefore x<0.

Solving inequalities:
inequalities-trick-91482.html (check this one first)
x2-4x-94661.html#p731476
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html?hilit=extreme#p873535
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.
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Re: If 4/x < 1/3 , what is the possible range of values of x? [#permalink]

25 Aug 2012, 03:50

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harshvinayak wrote:

Another way to solve this type of inequality:

From \frac{4}{x}<\frac{1}{3} it follows that x\neq0, therefore we can multiply both sides by 3x^2, which is positive.
We obtain 12x<x^2 or x^2-12x>0.
Now, imagine the graph of the quadratic function, y=12x^2-x, which is an upward parabola. See the attached drawing.
This parabola intercepts the X-axis at x=0 and x=12, the two "arms" are above the X-axis, meaning the values of y are positive when x<12 or x>0, and the values of y are negative (graph under the X-axis) when 0<x<12.

Hence the solution x<0 or x>12.

Note: Just remember the shape of the parabola, then you can easily deduce the sign of the quadratic function y=x^2+bx+c. If the quadratic equation x^2+bx+c=0 has two roots x_1<x_2 then y<0 (negative) between the two roots and y>0 (positive) outside the roots.
Or succinctly, y>0 if x_1<x<x_2 and y>0 if x<x_1 or x>x_2.

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ParabolaUp.jpg [ 8.97 KiB | Viewed 568 times ]

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Re: If 4/x < 1/3 , what is the possible range of values of x? [#permalink]

25 Aug 2012, 04:06

Ah!

thanks for the clarification Bunuel. I guess staying up all night took its toll. I had to reread and reread and reread and...

finally what I did was (per the first link you suggested) ::

dumping the two cases,
\frac{(12-x)}{(3x)} < 0
\frac{(x-12)}{(3x)} > 0 :idea:

roots: 12 and 0
and then I could ride the waves of + - + ...... weeeeeeeeeee.... weeeeeeeeeee... and weeeeeeeeeeee :-D

Thanks again for helping out Bunuel ..

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Re: If 4/x < 1/3 , what is the possible range of values of x? [#permalink]

25 Aug 2012, 04:15

Thanks EvaJager

you guys are awesome \m/

PS: I have just started out with my prep and have a lot to learn (and ask :lol:

) in little less than 3 weeks.
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Re: If 4/x < 1/3 , what is the possible range of values of x? [#permalink]

26 Aug 2012, 11:24

harshvinayak wrote:

Thanks EvaJager

you guys are awesome \m/

PS: I have just started out with my prep and have a lot to learn (and ask :lol:

) in little less than 3 weeks.

One way to understand this is that Sign of \frac{x-a}{x-b} is same as the sign of (x-a)(x-b)
and for (x-a)(x-b) always remember that it will be negative between a & b and positive outside a & b

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Re: If 4/x < 1/3 , what is the possible range of values of x? [#permalink]

28 Aug 2012, 21:58

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harshvinayak wrote:

Check out these posts. They discuss how to solve inequalities efficiently (using the wave) and how to handle various complications that can arise in a question.

http://www.veritasprep.com/blog/2012/06 ... e-factors/
http://www.veritasprep.com/blog/2012/07 ... ns-part-i/
http://www.veritasprep.com/blog/2012/07 ... s-part-ii/
http://www.veritasprep.com/blog/2012/07 ... qualities/
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Re: If 4/x < 1/3 , what is the possible range of values of x? [#permalink]

28 Aug 2012, 22:16

WOW !

Thanks Karishma, thats one awesome collection of inequalities stuff. A kudos alone wouldnt have cut it :-D

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Re: If 4/x < 1/3 , what is the possible range of values of x? [#permalink] 28 Aug 2012, 22:16

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If 4/x < 1/3 , what is the possible range of values of x?

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