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

Mylo O'Moore

Mylo O'Moore

Answered question

2021-03-18

Prove by contraposition that if x does not equal 5 and is irrational then 3xx5 is irrational.

Answer & Explanation

joshyoung05M

joshyoung05M

Skilled2021-03-19Added 97 answers

Suppose that 3x/(x-5) is rational, that is, 3xx5=q,
for some rational q. Then 3x=q(x5)=qx5q(3q)x=5qx=5q3q
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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