Prove that if x is rational and x≠0, then 1x is rational.

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Unpled · Accepted Answer

Step 1 Given: x is rational x≠0 TO PROOF: 1x is rational DIRECT PROOF Property rational numbers: If a is a rational number, then there exists two integers y and z such that a=yz(with z≠0). x is an rational number and thus there exists integers y and z such that: x=yz Since x¬0, we then also know that the numerator cannot be zero y≠0 We are interested in 1x :: 1x=1yz=zy Since z and y are integers (with y≠0), we then know that 1x is rational. B⊗ If x is rational and x≠0, then 1x is rational.

Vasquez · Accepted Answer

To prove that if x is rational and x≠0, then 1x is rational, we can start by assuming that x is a rational number. By definition, a rational number can be expressed as the ratio of two integers, where the denominator is not zero. So, we can write x as:x=pq where p and q are integers, and q≠0. Since x≠0, we know that p≠0 as well.Now, let&#039;s consider 1x. Using the definition of division, we have:1x=1pqTo divide by a fraction, we can multiply by its reciprocal. Therefore, we can rewrite the above expression as:1x=1·qpSimplifying further, we get:1x=qpHere, qp is the ratio of two integers, where p and q are integers and p≠0. Thus, 1x is also a rational number.Hence, we have proved that if x is rational and x≠0, then 1x is rational.In conclusion, we used the definition of rational numbers, along with the properties of division and multiplication, to show that if x is a non-zero rational number, then its reciprocal 1x is also a rational number.

RizerMix · Accepted Answer

Let&#039;s assume that x is a rational number. By definition, a rational number can be expressed as the ratio of two integers: x=ab, where a and b are integers and b≠0.We need to prove that 1x is also a rational number. To do that, we need to find integers c and d such that 1x=cd, where d≠0.To find c and d, we can take the reciprocal of x:1x=1ab=baSince a and b are integers and b≠0, ba is also a fraction with integers in the numerator and denominator. Therefore, we can express 1x as the ratio of two integers, which means 1x is a rational number.Hence, if x is a rational number (x≠0), then 1x is also a rational number.

nick1337 · Accepted Answer

Step 1:Now, we need to show that 1x is also a rational number. We can express 1x as 1pq. To simplify this expression, we can multiply the numerator and denominator by the reciprocal of pq, which is qp. Thus, we have:1x=1pq=1pq·qpqp=1·qppq·qpStep 2:Simplifying further, we get:1x=qp·qp=q2p2Since q2 and p2 are both integers (since q and p are integers), we can conclude that 1x is a rational number.Therefore, we have proven that if x is rational and x≠0, then 1x is also a rational number.

Prove that if x is rational and x\ne0, then 1/x is rationa

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