# Chapter 3 - Polynomial and Rational Functions - Exercise Set 3.6 - Page 420: 52

$(-∞, -4) ∪ (0, ∞)$ Consider the Rational Inequality as follows: $\frac{x+4}{x}\gt0$ 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. $\frac{x+4}{x}\gt0$ 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 $x=0$ This implies $x =0$ 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 interval. The boundary points divide the number line into three intervals: $(-∞, -4), (-4, 0), (0, ∞)$ Step 4. Now, one test value within each interval is chosen and $f$ is evaluated at that number. Intervals: $(-∞, -4), (-4, 0), (0, ∞)$ Test value: $-5$ $2$ $1$ Sign Change: Positive Negative Positive $f (x) > 0?$: 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 $(-∞, -4) or (0, ∞)$. Conclusion: Thus, the interval notation of the given inequality is $(-∞, -4) ∪ (0, ∞)$ and the graph of the solution set on a number line is shown as follows: 