Can we make the definition of irrationality measure ‘stronger’?

Question

By definition, $$0<\left\lvert \frac{p}{q}-x\right\rvert<\frac1{q^{\mu(x)-\epsilon}}$$ has infinitely many solutions $(p,q)$ for every $\epsilon>0$.

However, to prove a theorem in a current project, I would need the following inequality to have infinitely many solutions: $$ 0<\frac{p}{q}-x<\frac1{q^{\mu(x)-\epsilon}}$$ In other words, I need infinitely many good over-estimated rational approximations for $x$.

Since $x$ in my project is transcendental, I used $<0$ instead of $\le0$.

My questions are

1. Does the definition of irrationality measure directly implies the second inequality has infinitely many solutions?
2. If no, then what conditions can we add on $x$ such that the second inequality has infinitely many solutions? What are some examples of $x$ that satisfies the new conditions?
3. Or, indeed, is it impossible for the second inequality to have infinitely many solutions?

This question may lack context; I apologize for being unable to provide more.

Thanks in advance.