You want an interval such that $\displaystyle \left | f(x) - 6 \right | \leq 2$.
So you want $\displaystyle \left | x^2 - 2x - 3 \right | \leq 2$
So
$\displaystyle x^2 - 2x - 3 \leq 2$
and
$\displaystyle x^2 - 2x - 3 \geq -2$
For the first inequality solve:
$\displaystyle x^2 - 2x - 3 = 2$
$\displaystyle x^2 - 2x - 5 = 0$
So $\displaystyle x = 1 \pm \sqrt{6}$ by the quadratic formula. We want the "+" solution as the "-" solution is less than 0. So we want $\displaystyle x \leq 1 + \sqrt{6}$ to solve the inequality.
For the second inequality solve:
$\displaystyle x^2 - 2x - 3 = -2$
$\displaystyle x^2 - 2x - 1 = 0$
So $\displaystyle x = 1 \pm \sqrt{2}$ by the quadratic formula. We again want the "+" solution since the "-" solution is less than 0. So we want $\displaystyle x \geq 1 + \sqrt{2}$.
Putting the two together gives an interval for x:
$\displaystyle 1 + \sqrt{2} \leq x \leq 1 + \sqrt{6}$
-Dan