Answer:
A) The sum of <em>x</em> and <em>y</em> is a rational number.
Step-by-step explanation:
We will first prove that the given values of <em>x</em> and <em>y</em> are rational and then we will prove the fact that the sum of two rational numbers is a rational number.
A rational number is a number which is in the form
, where <em>p</em> and <em>q</em> are integers and <em>q</em> ≠ 0.
We can write - 5 as
and clearly 1 ≠ 0.
So, <em>x</em> = -5 is a rational number.
Similarly, <em>y</em> = -4 is also a rational number.
For the second part of the proof, consider any two rational numbers
and
.
Now,
![\frac{p}{q} +\frac{r}{s} =\frac{ps+rq}{qs}](https://tex.z-dn.net/?f=%5Cfrac%7Bp%7D%7Bq%7D%20%2B%5Cfrac%7Br%7D%7Bs%7D%20%3D%5Cfrac%7Bps%2Brq%7D%7Bqs%7D)
Since <em>p</em>, <em>q</em>, <em>r</em> and <em>s</em> are integers, <em>ps</em> + <em>rq</em> is also an integer.
Moreover, since <em>q</em> ≠ 0 and <em>s</em> ≠ 0, <em>qs</em> ≠ 0.
Therefore,
is also a rational number.
Hence, sum of two rational numbers is also a rational number.
Since <em>x</em> = -5 and <em>y</em> = -4 are rational numbers, <em>x</em> + <em>y</em> = (-5) + (-4) = -9 is also a rational number.