Lower bounds for the function $y =(e^{x-\alpha}-1)(x-\alpha)$ with $\alpha\in\mathbb{R}^+$ Consider the function $y =(e^{x-\alpha}-1)(x-\alpha)$ with $\alpha\in\mathbb{R}^+$. I need and estimate such as
$$(e^{x-\alpha}-1)(x-\alpha)\ge c (e^x-1)(x-\alpha)$$
or also
$$(e^{x-\alpha}-1)(x-\alpha)\ge c (e^{x-\alpha}-1)x$$
for some positive constants $c$.
I am in trouble to prove that at least one of these inequalities hold true.
Could someone please help me?
Thank you in advance!

Where is the function defined? If it is defined on $x\geq k$ for some $k>\alpha$, then the second inequality is equivalent to
$$\frac{x-\alpha}x\ge c$$
or
$$1-\frac \alpha x\geq c$$
which is clearly true.
Or if $x>0$, then in the first inequality, just $c=1$ does the job since it translates to showing
$$\frac{e^{x-\alpha}-1}{e^{x}-1}\geq 1$$
which is of course true since $e^{x-\alpha}=\frac {e^{x}}{e^\alpha}$
