Determining whether piecewise function y[n] = {x[n] if n even, -x[n] if n odd} is time invariant
Hint: You can re-write \$y[n]\$ as

$$y[n] = (-1)^{n}x[n]$$

which has only one case. \$(-1)^{n} = 1\$ if \$n\$ is even so \$y[n] = x[n]\$ if \$n\$ is even, and \$(-1)^{n} = -1\$ if \$n\$ is odd so \$y[n] = -x[n]\$ if \$n\$ is odd. Does that help?