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How many positive integer values of x will satisfy the inequality

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How many positive integer values of x will satisfy the inequality  [#permalink]

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New post 20 Oct 2019, 01:34
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How many positive integer values of x will satisfy the inequality (x - 7)(x - 9)(x - 11) . . . . . . . . (x - 99) < 0 ?

A. 23
B. 24
C. 29
D. 30
E. 31
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How many positive integer values of x will satisfy the inequality  [#permalink]

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New post 20 Oct 2019, 03:35
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Total no. of terms \(\frac{(Last term - first term)}{2}\)+1= \(\frac{(99-7)}{2}\) +1= 47 terms.

For the inequality to be true the product of 47 numbers should be negative, which requires odd number of negative numbers among 47 terms.

(i)For values of x from 1 to 6 all the 47 terms will be -ve.
{e.g. For x=1, (x - 7)(x - 9)(x - 11) . . . . . . . . (x - 99)= (-6)(-8)(-10) . . . . . . . . (-98)}

(ii)When x=10, there are 45 -ve terms.
{we can not take x=7, 8 , 9 For x= 7 and 9 inequality will equal to 0 and for x=8 we will get even no of -ve terms For x=8, (x - 7)(x - 9)(x - 11) . . . . . . . . (x - 99)= (1)(-1)(-3) . . . . . . . . (-91)}

When x=14, there are 43 -ve terms.
.
.
.
When x=98, there is 1 -ve term
{For x=98, (x - 7)(x - 9)(x - 11) . . . . . . . . (x - 99)= (98-7)(98-9)(98-11) . . . . . . . . (98-99)}

So total values of x from (ii) is from 10 to 98 with a difference of 4.

Total positive value possible for x = 6 + {\(\frac{(Last term - first term)}{4}\)+1}
= 6+{\(\frac{(98-10)}{4}\)+1}=29

Ans (C)
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How many positive integer values of x will satisfy the inequality  [#permalink]

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New post 15 Nov 2019, 00:57
we could use the wavy curve approach here let's say
(x - 7)(x - 9)(x - 11) . . . . . . . . (x - 99) < 0
its a series starting from 7,9,11.......99
tn=a+(n-1)d
99=7+(n-1)2
n=47
from the wavy curve approach, let us take range for 7-9-11-13
for the mentioned interval, we get 7<x<9 and 11<x<13
we arrive at two values for every 4 let's take it till 44th.. it sums up 22
22+1(till the 47th)
23+6(we haven't taken the first 6 values into consideration)
hence option C
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How many positive integer values of x will satisfy the inequality   [#permalink] 15 Nov 2019, 00:57
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