## College Algebra (6th Edition)

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