Properties of Inequalities I am unable to understand the following statements. If someone can explain with the help of numerical examples it would be great:


*

*If sides of an inequality are both positive or both negative, taking the reciprocal reverses the inequality

*If $0<x<1$ and $m$ and $n$ are integers with $m>n$, then $x^m<x^n<x$

*If $0<x<1$, then $\sqrt{x}>x$

*If $0<x<1$, then $\dfrac{1}{x}>x$ and $\dfrac{1}{x}>1$

Below are some numerical examples of each of these inequalities.


*

*You know that $2<4$. Now, $0.5=\dfrac{1}{2} > \dfrac{1}{4}=0.25$. Similarly, $-2>-4$. now, $-0.5=\dfrac{-1}{2}<\dfrac{-1}{4}=0.25$.

*Consider $a=4$ and $x=0.5$. Everything meets the stipulations. Then $xa=0.5(4)=2<4=a$.

*Suppose $x=\dfrac{1}{2}$ and $m=3$ and $n=2$. Then we have that $\left(\dfrac{1}{2}\right)^3 = \dfrac{1}{8}$. Furthermore, $\left(\dfrac{1}{2}\right)^2 = \dfrac{1}{4}$. We know that $\dfrac{1}{8}<\dfrac{1}{4}$.

*Suppose that $x=\dfrac{1}{4}$. Then we have that $\sqrt{\dfrac{1}{4}}=\dfrac{1}{2}$. Furthermore, we know that $\dfrac{1}{2}>\dfrac{1}{4}$.

*Suppose that $x=\dfrac{1}{2}$. Then we know that $\dfrac{1}{\left(\frac{1}{2}\right)}=2$. Furthermore, $2>\dfrac{1}{2}$. Also, $\dfrac{2}>1$.

