The number of possible integral values of the parameter #k# for which the inequality #k^2x^2 < (8k -3)(x+6)# holds true for all values of #x# satisfying #x^2 < x+2# is?

A) 0
B) 1
C) 2
D) 3

1 Answer
Cesareo R.
Nov 27, 2017

#0#

Explanation:

#x^2 < x + 2# is true for #x in (-1,2)#

now solving for #k#

#k^2 x^2 - (8 k - 3) (x + 6) < 0# we have

#k in ((24 + 4 x - sqrt[24^2 + 192 x - 2 x^2 - 3 x^3])/x^2, (24 + 4 x + sqrt[24^2 + 192 x - 2 x^2 - 3 x^3])/x^2)#

but

#(24 + 4 x + sqrt[24^2 + 192 x - 2 x^2 - 3 x^3])/x^2# is unbounded as #x# approaches #0# so the answer is #0# integer values for #k# obeying the two conditions.

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