Answer
$(1, 2)$
Work Step by Step
Consider the Rational Inequality as follows:
$\frac{x}{x-1}>2$
Here are the steps required for Solving Rational Inequalities:
Step 1: One side must be zero and the other side can have only one fraction, so we simplify the fractions if there is more than one fraction. Let us subtract 2 from both sides to obtain zero on the right.
$\frac{x}{x-1}>2$
$\frac{(x)}{x-1}-2>0$
$\frac{(x-2(x-1)}{x-1}>0$
$\frac{-x+2}{x-1}>0$
Step 2: Critical or Key Values are first evaluated. In order to this, set the numerator and denominator of the fraction equal to zero and then simplified rational inequality is solved.
$-x+2 = 0$
This implies
$x =2$
And
$x-1=0$
This implies
$ x =1$
These solutions are used as boundary points on a number line.
Step 3: Locate the boundary points on a number line found in Step 2 to divide the number line into intervals.
The boundary points divide the number line into three intervals:
$(-∞,1), (1, 2), (2, ∞)$
Step 4: Now, one test value within each interval is chosen and $f$ is evaluated at that number.
Intervals: $(-∞,1), (1, 2), (2, ∞)$
Test value: $0$ $1.5$ $3$
Sign Change: Negative Positive Negative
$f (x)> 0?$: F T F
Step 5: Write the solution set, selecting the interval or intervals that satisfy the given inequality. We are interested in solving $f (x)> 0$ . Based on our work done in Step 4, we see that $f (x)> 0$ for all x in $(1, 2)$ .
Conclusion: Thus, the interval notation of the given inequality is
$(1, 2)$ and the graph of the solution set on a number line is shown as follows:
