Chebychev's inequality

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I am having a bit of a problem with Chebychev's inequality which is:

[tex]P(|X-\mu |\geq \alpha )\leq \frac{\mathrm{Var}(X)}{\alpha ^2}[/tex]

For a positive [tex]\alpha[/tex]. Here X denotes a stochastic variable with mean [tex]\mu[/tex] and finite variance. I am asked to give a direct proof of this result, using the inequality

[tex]Z^2\geq \textbf{1}_{|Z|\geq 1}, \quad Z=\frac{X-\mu}{\alpha}[/tex].

I have solved this part of the problem already. Next, however, I am asked to give sufficient and necessary conditions for equality in Chebychev, and this is where I could use some help. My initial idea was to use the other inequality, but I am not quite sure how I can translate this into the Chebychev inequality. I figure that equality will be achieved for -1, 0, and 1 respectively, due to the nature of this characteristic function, but I should probably have further conditions on that. Any help would be appreciated.
 

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