Answer
$(-∞,-6]∪ or (-2, ∞)$
Work Step by Step
Consider the Rational Inequality as follows:
$\frac{x-2}{x+2}≤2$
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.
$\frac{x-2}{x+2}≤2$
$\frac{x-2}{x+2}-2≤0$
$\frac{-x-6}{x+2}≤0$
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 then simplified rational inequality is solved.
$-x-6 = 0$
This implies
$x =-6$
And
$x+2=0$
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 found in Step 2 to divide the number line into intervals.
The boundary points divide the number line into three intervals:
$(-∞,-6), (-6, -2), (-2, ∞)$
Step 4: Now, one test value within each interval is chosen and $f$ is evaluated at that number.
Intervals: $(-∞,-6), (-6, -2), (-2, ∞)$
Test value: $-7$ $-4$ $0$
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
$(-∞,-6)$ or $(-2, ∞)$
However, the inequality involves (less than or equal to), we must also include the solution of $f (x)=0$ , namely -6, in the solution set.
Conclusion: Thus, the interval notation of the given inequality is
$(-∞,-6]∪(-2, ∞)$ and the graph of the solution set on a number line is shown as follows:
