Answer
$(-∞, -3) ∪ (-1,1)$
Work Step by Step
Consider the Inequality as follows:
$\frac{1}{(x+1)}$$\gt$$\frac{2}{(x-1)}$
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.
Let us subtract right side from left side to obtain zero on the right.
$\frac{1}{(x+1)}$$\gt$$\frac{2}{(x-1)}$
$\frac{1}{(x+1)}$$-$$\frac{2}{(x-1)}$$\gt$$0$
$\frac{(-x-3)}{(x+1)(x-1)}$$\gt$$0$
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.
$-x-3=0$
This implies
$ x =-3$
and
$x+1 = 0$
This implies
$x = -1$
Also, $x-1 = 0$
This implies
$x = 1$
These solutions are used as boundary points on a number line.
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 four intervals:
$(-∞, -3), (-3, -1), (-1, 1), (1, ∞)$
4. Now, one test value within each interval is chosen and $f$ is evaluated at that number.
Intervals: $(-∞, -3), (-3, -1), (-1, 1), (1, ∞)$
Test value: $-4$ $-2$ $0$ $ 2$
Sign Change: Positive Negative Positive Negative
$f (x) > 0?$: T F T F
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
$(-∞, -3) $ or $(-1, 1)$ .
Conclusion: Thus, the interval notation of the given inequality is
$(-∞, -3) ∪ (-1, 1)$ and the graph of the solution set on a number line is shown as follows:
