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

To solve the inequality \(\left| \frac{x^2 + kx + 1}{x^2 + x + 1} \right| < 3\) for all real values of \(x\), we will follow these steps:

### Step 1: Rewrite the Inequality
We start with the inequality:
\[ \left| \frac{x^2 + kx + 1}{x^2 + x + 1} \right| < 3 \]
This can be rewritten as:
...