Here’s a summary of the conclusions for each pair of fractions.

\(\dfrac{4x}{7}\) and \(\dfrac{9}{14}\)

When \(x=\frac{9}{8}\) the two expressions are equal. When \(x<\frac{9}{8}\) then \(\frac{9}{14}\) is bigger. When \(x>\frac{9}{8}\) then \(\frac{4x}{7}\) is bigger.

\(\dfrac{5}{9}\) and \(\dfrac{2x}{12}\)

When \(x=\frac{10}{3}\) the two expressions are equal. When \(x<\frac{10}{3}\) then \(\frac{5}{9}\) is bigger. When \(x>\frac{10}{3}\) then \(\frac{2x}{12}\) is bigger.

\(\dfrac{3x}{4}+1\) and \(\dfrac{x}{4}+3\)

\(\frac{3x}{4}+1\) can be written as a single fraction: \(\frac{3x+4}{4}\). Similarly, \(\frac{x}{4}+3=\frac{x+12}{4}\).

The two expressions are equal when \(x=4\). When \(x<4\), \(\frac{x}{4}+3\) is bigger. When \(x>4\), \(\frac{3x}{4}+1\) is bigger.

\(\dfrac{8(1-x)}{5}\) and \(\dfrac{x}{6}\)

The two expressions are equal when \(x=\frac{48}{53}\). When \(x<\frac{48}{53}\) then \(\frac{8(1-x)}{5}\) is bigger. When \(x>\frac{48}{53}\) then \(\frac{x}{6}\) is bigger.

\(\dfrac{8}{2x}\) and \(\dfrac{4x}{16}\)

The two expressions are equal when \(x=-4\) or \(x=4\). When \(x>4\), \(\frac{x}{4}\) is bigger. When \(0<x<4\), \(\frac{8}{2x}\) is bigger. When \(-4<x<0\) then \(\frac{x}{4}\) is bigger. And finally, when \(x<-4\), \(\frac{8}{2x}\) is bigger.

When \(x=0\), \(\frac{8}{2x}\) is undefined so we are not able to compare the two fractions.