Given $y=\dfrac{x}{x-1}$ and $x\gt 1$, which of the following is a possible value of $y$ ?

$-1.9$

$-0.9$

$0.0$

$0.9$

$1.9$

Approach 1

Substitute each value in for $y$, and check if there is an $x$ value that could lead to the $y$ value. For example, checking the first choice.

$$ -1.9=\frac{x}{x-1} $$ $$ -1.9x+1.9=x $$ $$ 1.9=2.9x $$ $$ x= \frac{1.9}{2.9} $$

Which is not possible since $x\gt 1$.

Approach 2

Note that if $x\gt 1$, the fraction will always be a positive number over another positive number.

$$ \frac{x}{x-1} \gt 0 \text{ if } x\gt 1 $$

Next, note that the numerator will always be larger than the denominator, which means that the quotient will be greater than 1.