Answer
$(-∞, -5) ∪ (-3, ∞)$
Work Step by Step
Consider the Rational Inequality as follows:
$\frac{x+1}{x+3}<2$
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.
$\frac{x+1}{x+3}<2$
$\frac{(x+1)}{x+3}-2<0$
$\frac{-x-5}{x+3}<0$
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.
$-x-5 = 0$
This implies
$x =-5$
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 found in Step 2 to divide the number line into intervals.
The boundary points divide the number line into three intervals:
$(-∞, -5), (-5, -3), (-3, ∞)$
Step 4. Now, one test value within each interval is chosen and $f$ is evaluated at that number.
Intervals: $(-∞, -5), (-5, -3), (-3, ∞)$
Test value: $-6$ $-4$ $-2$
Sign Change: Negative Positive Negative
$f (x)< 0?$: 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 is
$(-∞, -5)$ or $(-3, ∞)$ .
Conclusion: Thus, the interval notation of the given inequality is
$(-∞, -5) ∪ (-3, ∞)$ and the graph of the solution set on a number line is shown as follows:
