Given a fraction with $ x $ is a real number, judge the scope of $ x $. From an ACT Math test:

Suppose that $ x $ is a real number and $ \frac { 4 x } { 6 x ^ 2 } $ is a rational number. Which of the following statements about $ x $ must be true?

*

*$ x $ is rational

*$ x $ is irrational

*$ x = 1 $

*$ x = \frac 2 3 $

*$ x = \frac 3 2 $

The answer says it must be a rational number, but how about an irrational number, say, $ \frac 4 3 $, which can also satisfy $ \frac { 4 x } { 6 x ^ 2 } $ a rational number?

Simply $\dfrac{4x}{6x^2}=\dfrac{4}{6x}$ can be irrational, but doesn't have to -- for example let's take $x=\dfrac{10}{3}.$
Then we have
$\dfrac{4}{6x}=\dfrac{2}{3x}=\dfrac{2}{3\cdot\frac{10}{3}}=\dfrac{2}{10}.$
It isn't irrational number definitely.
By the way the question says which statement must be true.
