To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $3\left(-2x+1\right)$ and $1-2x$ is $3\left(-2x+1\right)$. Multiply $\frac{6x}{1-2x}$ times $\frac{3}{3}$.
Use the distributive property to multiply $3$ by $-2x+1$.
$$\frac{5+18x}{-6x+3}\geq 0$$
For the quotient to be $≥0$, $18x+5$ and $3-6x$ have to be both $≤0$ or both $≥0$, and $3-6x$ cannot be zero. Consider the case when $18x+5\leq 0$ and $3-6x$ is negative.
$$18x+5\leq 0$$ $$3-6x<0$$
This is false for any $x$.
$$x\in \emptyset$$
Consider the case when $18x+5\geq 0$ and $3-6x$ is positive.
$$18x+5\geq 0$$ $$3-6x>0$$
The solution satisfying both inequalities is $x\in \left[-\frac{5}{18},\frac{1}{2}\right)$.
$$x\in [-\frac{5}{18},\frac{1}{2})$$
The final solution is the union of the obtained solutions.
