Many ways problem

## Main solutions

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.