Simplifying Inequality Involving $\sigma$, $\beta$ and $x$

Question

Given that $\sigma>0$, $\beta>0$, $x>0$ and $\sigma>\beta$, there are a couple of simplifications I cant derive: $$1.\,\,\sigma \geq x\,\,\,\,and\,\,\,\,\sigma\beta\geq x\,\,\,\,to\,\,\,\,x\leq\beta$$ $$2.\,\,\sigma \geq x\,\,\,\,and\,\,\,\,\sigma\beta< x\,\,\,\,to\,\,\,\,\beta< x\leq\sigma$$

I have tried using multiplication and division but cant figure out how to arrive to the final answer. Please help. Thanks

Answer 1

First one is not true: $\sigma = 2, \beta = \frac{1}{2}, x = 1.$

Second one is also not true: $\sigma = \frac{1}{2}, \beta = \frac{2}{5}, x = \frac{3}{10}.$