Problem 52 - Algebra and Trigonometry

Question

Solve each rational inequality in Exercises \(43-60\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+4}{x}>0 $$

Step-by-Step Solution

Answer

The solution set for the inequality \( \frac{x+4}{x} > 0 \) in interval notation is \( (-\infty, -4) \cup (0, +\infty) \).

Step 1: Determine the Critical Values

Critical values are those values that make the rational inequality equal to zero or undefined. In this case, the critical values for the inequality \( \frac{x+4}{x}> 0 \) are -4 (where the numerator equals to zero) and 0 (where the denominator equals to zero).

Step 2: Test the Critical and Non-Critical Intervals

The critical values split the number line into intervals: \(-\infty , -4\), \(-4, 0\), and \(0, +\infty\). Choose representatives in each one of these intervals and substitute back into the inequality to check where it holds. As an example, let's choose -5 for the interval \(-\infty , -4\), -1 for \(-4, 0\), and 1 for \(0, +\infty\). After testing these values, we see that the inequality holds for intervals \(-\infty , -4\) and \(0, +\infty\).

Step 3: Express the solution in Interval Notation and Graph

Now that we have determined the solution set of the inequality, it can be expressed in interval notation as \((-\infty, -4) \cup (0, +\infty)\). This is also the range of solutions to our initial problem. Graphing this solution would involve graphing these intervals on a real number line, showing an open circle at -4 and 0 and shading to the left of -4 and to the right of 0.