The Erdős–Straus conjecture - unresolved root goal

For every natural number $n$ with $2<n$, there exist natural numbers $x$, $y$, and $z$ such that $1\le x$, $x<y$, and $y<z$, and $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$. This equality is an equality of rational numbers, with each natural-number denominator interpreted as a rational number. The inequalities ensure that all denominators are nonzero and that $x$, $y$, and $z$ are distinct and positive. No conclusion is asserted for $n=0$, $n=1$, or $n=2$.