Standard Normal Distribution-Inequality

Question

Suppose that $\Phi(.)$ and $\phi(.)$ are the cdf and the pdf of the standard Normal distribution, respectively. Does the following inequality hold $\forall x>0$:
$$2x\phi(x)+2\Phi(x)-1\geq 0$$

Answer 1

For $x>0$ we have $\Phi(x)>\frac12$, hence $$2\Phi(x)-1 \geq 0 $$ and since $x\phi(x)$ is positive for positive $x$ it clearly holds.