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Manager  Status: Still Struggling
Joined: 02 Nov 2010
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GMAT Date: 10-15-2011
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WE: Information Technology (Computer Software)
(x^2 - |x| - 12)/(x-3) >= 2x  [#permalink]

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Hi Guys,

$$\frac{x^2 - |x| - 12}{x-3} \geq 2x$$
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 10226
Location: Pune, India
Re: (x^2 - |x| - 12)/(x-3) >= 2x  [#permalink]

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krishnasty wrote:
Hi Guys,

x^2 - |x| - 12
--------------- >= 2x
x-3

---------------------------------------------
Consider KUDOS if you like my post!!!

I am assuming your question is: $$\frac{{x^2 - |x| - 12}}{{x - 3}} >= 2x$$
First of all, remember that we cannot cross multiply in this case. The reason is that we do not know the sign of (x - 3). To handle the 2x on the right hand side, bring it to the left hand side to get:

$$\frac{{x^2 - |x| - 12}}{{x - 3}} - 2x >= 0$$

$$\frac{{-x^2 - |x| + 6x - 12}}{{x - 3}} >= 0$$

Now take 2 cases.

Case 1: If x >= 0, |x| = x
$$\frac{{-x^2 - x + 6x - 12}}{{x - 3}} >= 0$$

$$\frac{{x^2 - 5x + 12}}{{x - 3}} <= 0$$ (We flipped the sign of left hand side and hence the inequality flipped too.)

Now, $$x^2 - 5x + 12$$ has no real roots. It will always be positive. (because $$b^2 - 4ac$$ is negative so we know that it has no real roots. Put x = 0, you get 12 which is positive. So this quadratic will never touch the x axis and will always remain above it.)

So for the left hand side to be negative, x < 3. (x cannot be 3 because in that case the denominator will be 0.)
Since x has to be greater than 0, the only solution here is 0 <= x < 3.

Case 2: If x <= 0, |x| = -x
$$\frac{{-x^2 + x + 6x - 12}}{{x - 3}} >= 0$$

$$\frac{{x^2 - 7x + 12}}{{x - 3}} <= 0$$ (We flipped the sign of left hand side and hence the inequality flipped too.)

$$\frac{{(x - 4)(x - 3)}}{{x - 3}} <= 0$$
or x <= 4
Since x has to be less than(or equal to) 0, the only solution here is x <= 0.

Combining the solutions above, we get x < 3 as the final solution.

(I have assumed that you are comfortable with quadratic expressions and solving for inequalities once you have factors. If you are not, let me know and I will point you to the related theory.)
_________________
Karishma
Veritas Prep GMAT Instructor

##### General Discussion
Intern  Joined: 27 Aug 2010
Posts: 18
Re: (x^2 - |x| - 12)/(x-3) >= 2x  [#permalink]

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VeritasPrepKarishma wrote:
krishnasty wrote:
Hi Guys,

x^2 - |x| - 12
--------------- >= 2x
x-3

---------------------------------------------
Consider KUDOS if you like my post!!!

I am assuming your question is: $$\frac{{x^2 - |x| - 12}}{{x - 3}} >= 2x$$
First of all, remember that we cannot cross multiply in this case. The reason is that we do not know the sign of (x - 3). To handle the 2x on the right hand side, bring it to the left hand side to get:

$$\frac{{x^2 - |x| - 12}}{{x - 3}} - 2x >= 0$$

$$\frac{{-x^2 - |x| + 6x - 12}}{{x - 3}} >= 0$$

Now take 2 cases.

Case 1: If x >= 0, |x| = x
$$\frac{{-x^2 - x + 6x - 12}}{{x - 3}} >= 0$$

$$\frac{{x^2 - 5x + 12}}{{x - 3}} <= 0$$ (We flipped the sign of left hand side and hence the inequality flipped too.)

Now, $$x^2 - 5x + 12$$ has no real roots. It will always be positive. (because $$b^2 - 4ac$$ is negative so we know that it has no real roots. Put x = 0, you get 12 which is positive. So this quadratic will never touch the x axis and will always remain above it.)

So for the left hand side to be negative, x < 3. (x cannot be 3 because in that case the denominator will be 0.)
Since x has to be greater than 0, the only solution here is 0 <= x < 3.

Case 2: If x <= 0, |x| = -x
$$\frac{{-x^2 + x + 6x - 12}}{{x - 3}} >= 0$$

$$\frac{{x^2 - 7x + 12}}{{x - 3}} <= 0$$ (We flipped the sign of left hand side and hence the inequality flipped too.)

$$\frac{{(x - 4)(x - 3)}}{{x - 3}} <= 0$$
or x <= 4
Since x has to be less than(or equal to) 0, the only solution here is x <= 0.

Combining the solutions above, we get x < 3 as the final solution.

(I have assumed that you are comfortable with quadratic expressions and solving for inequalities once you have factors. If you are not, let me know and I will point you to the related theory.)

+1 to you .
Thats a Good 1 Senior Manager  Joined: 21 Dec 2010
Posts: 325
Re: (x^2 - |x| - 12)/(x-3) >= 2x  [#permalink]

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okies, this is a great explanation , concept cleared , thanks karishma. there is one question : when can one be sure that a quadratic expression is always +ve ?
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 10226
Location: Pune, India
Re: (x^2 - |x| - 12)/(x-3) >= 2x  [#permalink]

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garimavyas wrote:
okies, this is a great explanation , concept cleared , thanks karishma. there is one question : when can one be sure that a quadratic expression is always +ve ?

When you see x^2 - 5x + 12, think - will it have roots such that their sum is -5 and product is 12? Since product is +ve, and sum -ve, both roots should be negative. But will any two -ve roots add up to give -5 such that their product is 12? (e.g. -1, -4 or -2, -3 etc we do not get 12 as product) No.

To confirm, notice that b^2 - 4ac (where a = 1, b = -5 and c = 12) is -ve here so it has no real roots. Just put x = 0 and you get x^2 - 5x + 12 = 12 which is positive. This means the graph of this expression lies above the x axis. Remember, it will not meet the x axis anywhere because it has no real roots. Hence x^2 - 5x + 12 will always be positive.
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Karishma
Veritas Prep GMAT Instructor

Senior Manager  Joined: 21 Dec 2010
Posts: 325
Re: (x^2 - |x| - 12)/(x-3) >= 2x  [#permalink]

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hey ,
clearer now, thanks a ton
Math Expert V
Joined: 02 Sep 2009
Posts: 62351
Re: (x^2 - |x| - 12)/(x-3) >= 2x  [#permalink]

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garimavyas wrote:
okies, this is a great explanation , concept cleared , thanks karishma. there is one question : when can one be sure that a quadratic expression is always +ve ?

Quadratic expression $$ax^2+bx+c$$ is always positive when 1. $$a$$, the coefficient of $$x^2$$, is positive: $$a>0$$ AND 2. the equation $$ax^2+bx+c=0$$ has no real roots, which happens when discriminant is negative, so when $$d=b^2-4ac<0$$.

In this case the graph (parabola) of the function $$f(x)=ax^2+bx+c$$ is upward and lies above the X-axis, thus is always positive.

Similarly quadratic expression $$ax^2+bx+c$$ is always negative when 1. $$a$$, the coefficient of $$x^2$$, is negative: $$a<0$$ AND 2. the equation $$ax^2+bx+c=0$$ has no real roots, which happens when discriminant is negative, so when $$d=b^2-4ac<0$$.

Check Coordinate Geometry chapter of Math Book for more: math-coordinate-geometry-87652.html

Hope it helps.
_________________
Senior Manager  Joined: 21 Dec 2010
Posts: 325
Re: (x^2 - |x| - 12)/(x-3) >= 2x  [#permalink]

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that solidifies the work of Karishma . now it is formulated .

thanks Karishma and Bunuel
Non-Human User Joined: 09 Sep 2013
Posts: 14447
Re: (x^2 - |x| - 12)/(x-3) >= 2x  [#permalink]

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_________________ Re: (x^2 - |x| - 12)/(x-3) >= 2x   [#permalink] 12 Aug 2018, 03:28
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