## College Algebra (6th Edition)

$(-∞,-6]∪ or (-2, ∞)$
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: