Problem 45 - 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. $$ \frac{x+3}{x+4}<0 $$

Step-by-Step Solution

Answer

The solution to the inequality is \((-\infty, -4) \cup (-3, \infty)\).

Step 1: Identify the critical values

The denominator cannot be 0, therefore \(x \neq -4\). Also, the inequality will be 0 when the numerator is 0, i.e., when \(x = -3\). Therefore, the critical values for this problem are \(x = -4\) and \(x = -3\)

Step 2: Test intervals

The critical values divide the number line into three regions: \(-\infty$ to \(-4\), \(-4\) to \(-3\), and \(-3\) to \(\infty\). To test which interval solves the inequality, choose a sample point in each interval and substitute it into the original inequality. If it makes the inequality true, then the interval is part of the solution, otherwise it is not. The three test points are \(x = -5\), \(x = -3.5\), and \(x = 0\)

Step 3: Solve for each interval

Substitute each test point. For \(x = -5\), the inequality becomes \(\(-2/ -1 < 0\) which is false. For \(x = -3.5\), the inequality becomes \(\( -0.5/ -0.5 < 0\) which is also false. For \(x = 0\), the inequality becomes \(3/ 4 < 0\) which is false. The solutions are the intervals where the inequality is not true, so in this case, the solutions are \(-\infty$ to \(-4\) (excluding \(-4\)) and \(-3\) to \(\infty\).

Step 4: Express solutions in interval notation

The solution set in interval notation is \((-\infty, -4) \cup (-3, \infty)\).