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# Not able to derive this equation..

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Intern
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Not able to derive this equation.. [#permalink]

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12 Sep 2012, 20:10
(10-w)(9-w) < 9 ... implies w > 6 ...how?
w(w-1) > 45 ..implies w>7...how?

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Re: Not able to derive this equation.. [#permalink]

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13 Sep 2012, 23:32
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gbisnik wrote:
(10-w)(9-w) < 9 ... implies w > 6 ...how?
w(w-1) > 45 ..implies w>7...how?

I think it would be given to you that w is an integer.

If this is a part of a GMAT question, rest assured you would not have to deal with it mathematically. There would be no time to do that.
As shown above, you will get w^2 -19w + 81 < 0 and the apprx roots as 6.5 ans 12.5 which will give (w - 6.5)(w - 12.5) < 0 which implies 6.5 < w < 12.5
(Check out this post if you have doubts in the last step: http://www.veritasprep.com/blog/2012/06 ... e-factors/ )

Instead, you only need to deal with it logically.

w(w-1) > 45
Think of the case where this is an equation: w(w-1) = 45
Two numbers which are close to each other give 45. We know 7^2 = 49 which is close to 45. If we put w = 7, we get 7*6 = 42 which is less than 45. So w must be greater than 7 for the product to be greater than 45.

Similarly, (10-w)(9-w) < 9
If the product of two numbers close to each other is 9, the number must be around 3. If w = 6, then the numbers are 4 and 3. But 4*3 = 12 which is greater than 9. Hence, w must be more than 6 to get the numbers smaller than 4 and 3 and hence the product less than 9.
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Re: Not able to derive this equation.. [#permalink]

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12 Sep 2012, 20:49
gbisnik wrote:
(10-w)(9-w) < 9 ... implies w > 6 ...how?
w(w-1) > 45 ..implies w>7...how?

(10-W)(9-W)<9 just do trial and error method for solving this problem. you will find that if W>6 only it gives less than 3.

and in the similar fashion for 2nd question also.

If you like please give +1 kudos
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Re: Not able to derive this equation.. [#permalink]

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13 Sep 2012, 01:50
(10-w)(9-w) < 9 ... implies w > 6 ...how?
w(w-1) > 45 ..implies w>7...how?

(10-w)(9-w) < 9
expand
90 -19w + w^2 < 9

w^2 -19w + 81 < 0
roots will be close to 6.5 and 12.5
(w-6.5) * (w-12.5) < 0

So either (w-6.5) < 0 and (w-12.5) > 0
or (w-6.5) > 0 and (w-12.5) < 0

(w-6.5) < 0 and (w-12.5) > 0
=> w <6.5 and w > 12.5 NO intersection between these two so no solution

(w-6.5) > 0 and (w-12.5) < 0
=> w > 6.5 and w< 12.5
=> 6.5 < w < 12.5
So if w is integer then 7 <= w <= 12

w(w-1) > 45
expand

w^2 - w - 45 > 0
roots will be close to 7.25 and -6.25
So, (w - 7.25) * (w + 6.25) > 0
So either both the terms are > 0 or both are < 0

(w - 7.25) < 0 and (w + 6.25) < 0
=> w < 7.25 and w < -6.25 Intersection is w < -6.25

(w - 7.25) > 0 and (w + 6.25) > 0
=> w > 7.25 and w > -6.25
Intersection is w > 7.25
So, answer for second equation is w < -6.25 and w > 7.25

Hope it helps!
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Re: Not able to derive this equation.. [#permalink]

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13 Sep 2012, 02:28
gbisnik wrote:
(10-w)(9-w) < 9 ... implies w > 6 ...how?
w(w-1) > 45 ..implies w>7...how?

This is an inequation and not an equation (equation means equality between two algebraic expressions).
An inequation, usually has infinitely many solutions.

You can rewrite $$(10-w)(9-w)<9$$ as $$w^2-19w+81<0$$.
The equation $$y=w^2-19w+81$$ is the equation of an upward parabola, which intersects the horizontal axis at the roots of the equation $$y=0.$$ Sketch the graph of the parabola after you find/estimate the roots, then read from the graph the values for which the inequality holds. For an upward parabola, the expression is negative for the values of the variable between the two roots and positive for values less than the smallest root or greater than the largest root.
In this case, the roots are pretty ugly ($$\frac{19\pm\sqrt{37}}{2}$$), approximately 12.5 and 6.45, so the given expression is negative for $$w$$ between the previous two values. If $$w$$ is an integer, then it cannot be greater than 6.

Proceed similarly with the second inequality.

Good luck!
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Re: Not able to derive this equation..   [#permalink] 13 Sep 2012, 02:28
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