Answer
$(-∞, 0) ∪ (3, ∞)$
Work Step by Step
Consider the Rational Inequality as follows:
$\frac{x}{x-3}\gt0$
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.
$\frac{x}{x-3}\gt0$
Step 2: Critical or Key Values are first evaluated. In order to do this, set the numerator and denominator of the fraction equal to zero and solve.
$x = 0$
This implies
$x =0$
and
$x-3=0$
This implies
$ x =3$
These solutions are used as boundary points on a number line.
Step 3: Locate the boundary points on a number line and divide the number line into intervals.
The boundary points divide the number line into three intervals:
$(-∞, 0), (0,3), (3, ∞)$
Step 4. Now, one test value within each interval is chosen and $f$ is evaluated at that number.
Intervals: $(-∞, 0), (0,3), (3, ∞)$
Test value: $-1$ $2$ $4$
Sign Change: Positive Negative Positive
$f (x) > 0?$: T F T
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
$(-∞, 0) or (3, ∞)$.
Conclusion: Thus, the interval notation of the given inequality is
$(-∞, 0) ∪ (3, ∞)$ and the graph of the solution set on a number line is shown as follows:
