Answer
$(-∞, -1) ∪ (1, 2) ∪ (3, ∞)$
Work Step by Step
Consider the Inequality as follows:
$\frac{x^{2} -3x +2}{x^{2} -2x -3}$$> 0$
Here are the steps required for Solving Rational Inequalities:
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.
$x^{2} -3x +2 = 0$
This implies
$ x =1,2$
and
$x^{2} -2x -3 = 0$
This implies
$x = -1, 3$
These solutions are used as boundary points on a number line.
2. Locate these boundary points on a number line found in Step 1 to divide the number line into intervals.
The boundary points divide the number line into five intervals:
$(-∞, -1), (-1, 1), (1, 2), (2, 3), (3, ∞)$
3. Now, one test value within each interval is chosen and $f$ is evaluated at that number.
Intervals: $(-∞, -1) (-1, 1) (1, 2) (2, 3) (3, ∞)$
Test value: $-2$ $0$ $1.5$ $ 2.5$ $4$
Sign Change: Positive Negative Positive Negative Positive
$f (x) > 0?$: T F T F T
4. Write the solution set, selecting the interval or intervals that satisfy the given inequality.
Based on our work done in Step 3, we see that $f (x) > 0$ for all $x$ in $(-∞, -1)$or $(1, 2)$ or$(3, ∞)$.
Thus, the interval notation of the given inequality is $(-∞, -1) ∪ (1, 2) ∪ (3, ∞)$ and the graph of the solution set on a number line is shown as follows: