quadratic equation

[kinu]

the number of integral values of \(k\) for which the equation \(x^2-4\mid x\mid +3-\mid k-1\mid =0\)

has at least \(3\) distinct real roots

[João P. Ferreira]

Hi remind that

\(|x| = \begin{cases} x, & \mbox{if } x \ge 0 \\ -x, & \mbox{if } x < 0. \end{cases}\)

and

\(|k-1| = \begin{cases} k-1, & \mbox{if } k \ge 1 \\ -k+1, & \mbox{if } k < 1. \end{cases}\)

Then just decompose in four options and try to find those which give 3 real roots...