Suppose that 3x/(x-5) is rational, that is, \(\displaystyle{3}\frac{{x}}{{{x}-{5}}}={q}\),

for some rational q. Then \(\displaystyle{3}{x}={q}{\left({x}-{5}\right)}={q}{x}-{5}{q}\to{\left({3}-{q}\right)}{x}=-{5}{q}\to{x}=-{5}\frac{{q}}{{{3}-{q}}}\)

Since q is rational, then 3-q and -5q are also rational, and their quotient is also rational, meaning that x is rational.

By contraposition, we have proved the required statement.

for some rational q. Then \(\displaystyle{3}{x}={q}{\left({x}-{5}\right)}={q}{x}-{5}{q}\to{\left({3}-{q}\right)}{x}=-{5}{q}\to{x}=-{5}\frac{{q}}{{{3}-{q}}}\)

Since q is rational, then 3-q and -5q are also rational, and their quotient is also rational, meaning that x is rational.

By contraposition, we have proved the required statement.