Answer
$(-∞, -4] ∪ (-2, 1]$
Work Step by Step
Consider the Rational Inequality as follows:
$\frac{(x+4)(x-1)}{x+2}≤0$
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.
$\frac{(x+4)(x-1)}{x+2}≤0$
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 solve.
$x+4 = 0$
This implies
$x =-4$
and
$x-1=0$
This implies
$ x =1$
Also, $x+2=0$
This implies
$ x =-2$
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 four intervals:
$(-∞, -4), (-4, -2),(-2, 1), (1, ∞)$
Step 4: Now, one test value within each interval is chosen and $f$ is evaluated at that number.
Intervals: $(-∞, -4), (-4, -2), (-2, 1), (1, ∞)$
Test value: $-5$ $-3$ $0$ $2$
Sign Change: Negative Positive Negative Positive
$f (x)< 0?$: T 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
$(-∞, -4]$ or $(-2, 1]$ .
However, the inequality involves (less than or equal to), we must also include the solution of $f (x)= 0$ , namely -4 and 1 in the solution set.
Conclusion: Thus, the interval notation of the given inequality is $(-∞, -4] ∪ (-2, 1]$ and the graph of the solution set on a number line is shown as follows:
