Find the values of `k` for which `|(x^2+k x+1)/(x^2+x+1)|<2,AAx in R`

Explanation

We have
`|x| lt a rArr - a lt x lt + a`
`therefore` The given inequality implies
`-2lt(x^(2) + kx + 1)/(x^(2) + x + 1) lt 2" "...(1)`
Now, `x^(2) + x +1 = (x+ 1/2)^(2) + (3/4)` is `+`ve for all values of `x`.
Multiplying `(1)` by `x^(2) + x +1,` we get
` -2(x^(2) + x+1) lt x^(2) + kx + 1 lt 2(x^(2) + x + 1)`
This yields two inequalities, viz.,
`3x^(2) + (2+k)x + 3) gt`0
and `x^(2) + (2-k) x + 1 lt 0`
For above inequalities to be. true for all values of x, their discriminants must be negative. Hence,
`(2+k)^(2) - 36 lt 0` and `(2 - k)^(2) - 4 lt 0`
`rArr (k+ 8) (k - 4) lt 0 and k (k - 4) lt 0`
`rArr - 8 lt k lt 4 and 0 lt k lt 4`
Therefore, `0lt k lt 4`