If x is a positive integer such that (x-1)(x-3)(x-5)....(x-93) < 0, ho

Let me explain this in much more easier way
We do this question by plugging method
Here no. of terms are 47
X is a positive integer so,
Let x=1
so 0 is not less than 0
let x=2
1* rest negative terms are even so result will be >0
Let x=3
2*0*....=0
Let x=4
3*1* rest 45 terms are negative so answer is <0
therefore this is true for x=4 and its multiples
until 92

If x is a positive integer such that (x-1)(x-3)(x-5)....(x-93) < 0, how many values can x take

If x >= 94, then the overall product is positive. If x is an odd number less than 94, then the overall product is 0. Therefore, given that x must be a non-odd positive integer less than 94, x must be an even number.

If x = 2, then (x-1) is positive and the other factors are negative, so what matters is how many remaining factors e.g. (x - 3), (x - 5) are negative.

2n - 1 = 93
2n = 94
n = 47

There are 47 factors. If (x - 1) is positive, then 47 - 1 = 46 factors are negative, and when you multiply an even number of negative numbers with a positive number, you get a positive number as a result. Therefore, 2 can't be a solution.

For the next even number, 4, (x-1) and (x-3) are positive, leaving 47 - 2 = 45 negative factors. Since an odd number of negative numbers multiplied with a positive number results in a negative number, 4 is a solution.

It's clear that for each even x, the left hand remains positive while the product of the remaining factors alternates between positive and negative. Therefore, half of the even numbers below 47 are solutions. (47-1)/2 = 23.

There are 23 solutions.

Posted from my mobile device

Solution:

METHOD 1 (logic)

• Firstly, value of x has to be less than 93. Because if lets say \(x=94\), then all the terms x-1, x-3, x-5, ...., x-93 will be positive and their product will also be positive
• Secondly, values of x cannot be odd. Because if lets say \(x=5\), then \(x-5=0\) and the whole product will become 0
• So, x is even less than 93 but not all even numbers
• x cannot be even numbers like 2, 6, 10 etc because these would make odd numbers of terms negative and thus the whole product positive
  
  • Since there are a total 47 (odd) number of terms, therefore for the product to be negative, we will need even number of terms to be negative
• The possible values of x are 4, 8, 12....92 which is an AP
• We can write \(92=4+(n-1)4\) or \(n=23\)

METHOD 2 (methodical and number line)

• Upon plugging the zero points and marking positive and negative ranges alternately, we get
  
  Attachment:
  
  negative.png [ 3.07 KiB | Viewed 2580 times ]
  
• We can clearly see that the possible values of x are 4, 8, 12....92 which is an AP
• We can write \(92=4+(n-1)4\) or \(n=23\)

Hence the right answer is Option C