inequality, JMO

How many positive integer values can x take that satisfy the inequality 
(x - 8) (x - 10) (x - 12).......(x - 50) < 0




Answer:

(x - 8) (x - 10) (x - 12).......(x - 50) < 0

No. of terms = (50 – 8)/2 + 1 = 22

If X < 8, then 22 –ve terms, which makes it +ve. Inequality does not hold.

If 8 < x < 10, i.e. x = 9, we have 21 –ve terms and 1 +ve term; the inequality holds

Similarly if 10 < x < 12, i.e. x = 11, we have 20 –ve terms and 2 +ve term; the inequality does not hold

We, find that for every alternate odd number starting from 9 the inequality holds.

So, the numbers are 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49

We have 11 positive integers for which the inequality will hold.