Put the steps in correct order to prove that if x is irrational, then 1/x is irrational using contraposition. a. Substitute 1/x = p/q, to obtain x = 1/(1/x) = 1/(p/q) = q/p.
b. Hence x is rational.
c. If 1/x is rational, then 1/x = p/q for some integers p and q with q ≠ 0. Observe that p is not 0 either, because 1/x is not 0.
d. Observe that x is the quotient of two integers with the denominator nonzero.

The correct order is:

a

c

d

b

Step-by-step explanation:

First, let's write 1/x in a convenient way for us:

a) Substitute 1/x = p/q, to obtain x = 1/(1/x) = 1/(p/q) = q/p.

Now we assume that 1/x is rational (we want to prove that this implies that x will be also rational and because we know that x is irrational assuming that 1/x is rational will lead to an incongruence), then:

c. If 1/x is rational, then 1/x = p/q for some integers p and q with q ≠ 0. Observe that p is not 0 either, because 1/x is not 0.

Now we know that we can write x as a quotient of two integers, we need to imply that, then the next one is:

d) Observe that x is the quotient of two integers with the denominator nonzero.

And that is the definition of rational, then we end with:

b) Hence x is rational.

Which is what we wanted to get.

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