Prove by contraposition that if x does not equal 5 and is irrational then 3x/x-5 is irrational.

Question
Equations and inequalities
asked 2021-03-18
Prove by contraposition that if x does not equal 5 and is irrational then \(\displaystyle{3}\frac{{x}}{{x}}-{5}\) is irrational.

Answers (1)

2021-03-19
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.
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