Prove that if $\Re(z)>x$ then $|z|>|x|$

Question

Let $x\in\Bbb{R}, z\in \Bbb{C}$. I wish to prove that if $\Re(z)>x$ then $|z|>|x|$. My idea was as follows: Let $z=a+bi$

$$|z|=\sqrt{a^2+b^2}\geq \sqrt{a^2}=|a|=|\Re(z)|\geq\Re(z)>x$$ But i couldn't somehow get the $|x|$ into the inequalities. If $x>0$ we have that $x=|x|$ and I'm done but what if $x$ is negative? would someone help please?

Answer 1

The statement is false. Take $z=0$ and $x=-1$. Then $\operatorname{Re}z>x$, but it is not true that $\lvert z\rvert>\lvert x\rvert$.