The number of integral values of K' the inequality `|(x^2+kx+1)/(x^2+x+1)|<3` is satisfied for all real values of x is....

Explanation

We have `|(x^(2)+kx+1)/(x^(2)+x+1)|lt3`
`implies-3lt(x^(2)+kx+1)/(x^(2)+x+1)lt3`
Since `x^(2)+x+1=(x+1/2)^(2)+3/4gt0`
`:.-3(x^(2)+x+1)ltx^(2)+kx+1lt3(x^(2)+x+1)`
`:.4x^(2)+(k+3)x+4gt0`……..i
and `2x^(2)-(k-3)x+2gt0` ……….ii
`:'4gt0` and `2gt0`
The inequality (i) will be valid, if ltbr `(k+3)^(2)-4.4.4lt0implies(k+3)^(2)lt64`
or `-8ltk+3lt8`
or `-11ltklt5`.........iii
and the inequality (ii) will be valid if
`(k-3)^(2)-4.2.2lt 0` or `(k-3)^(2)lt16`
or `-11ltklt7`.........iv
The conditions (iii) and (iv) will hold simultaneously, if
`-1ltklt5`