Determine if a statement is true or false

Question

I need to determine if this statement is true or false: For each non-zero integer x, there is a negative rational number y such that $x = \frac{3}{6y+4}$.

If I say x is rational it can be written as x=a/b and the same for y, so y=c/d

then I get (c/d)=(d/2c)-(a/b)

what do I do after this and how do I know if the statement is true or false?

Answer 1

You are on the right track, you correctly found out that if such an $r$ exists, it must be

$$r=\frac1{2m}-\frac23.$$

Even the converse is true: If you use this $r$, then $m=\frac3{6r+4}$ holds. The only tricky thing you need to remember here is that you need to check that the denominator $6r+4$ is always unequal to zero, but that works out.

Now, $m$ is supposed to be a non-zero integer between $-4$ and $4$. That leaves 8 possible values, so you could put them into your forumula one by one and see if the result is always a negative rational number. That's some work, but not too much, as the formula is simple.

Of course, it's often better to use some math insight to avoid too much 'legwork'. The first insight is that the $r$ calculated above will always be a rational number, so the only problem is finding out if it is negative.

If $m < 0$, then we have $\frac1m < 0$ as well, and thus $\frac1{2m} < 0$ and finally $r=\frac1{2m} -\frac23< 0$. So for all negative $m$, $r$ is negative as well (we didn't even need to use the fact that $m$ was an integer in that case).

If $m > 0$, then we know that $m \ge 1$, as $m$ is an integer. The function $f(x)=\frac1x$ is monotonically decreasing for positive real $x$ (draw a graph of that function to see that, or start from $0 < x < y$ and divide that by the positive number $xy$ to arrive that $0 < f(y)=\frac1y < \frac1x=f(x)$).

That means for positive $m$ we get

$$r=\frac1{2m}-\frac23=\frac12\frac1m-\frac23 \le \frac12\frac11-\frac23=\frac12-\frac23=-\frac16 < 0.$$

So also for positive $m\ge1$, we find that $r$ is negative.

Note that we now know that $r$ is negative for all non-zero integers $m$, not just the ones between $-4$ and $4$. We didn't have to use $m \ge -4$ or $m \le 4$ in the proof!