Problem 53 - 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. $$ \frac{(x+4)(x-1)}{x+2} \leq 0 $$

Step-by-Step Solution

Answer

The solution to the inequality is \[-4, -2) \cup [1, ∞)\]

Step 1: Find the critical points

To find critical points, set both numerator and denominator of the rational inequality equal to zero and solve for x. We get critical points as \(x = -4\), \(x = 1\) and \(x = -2\).

Step 2: Plot the critical points on the number line

The critical points plotted on the number line divide it into four intervals which are \((-∞, -4)\), \((-4, -2)\), \((-2, 1)\), and \( (1, ∞)\). Note that -2 is undefined so we exclude it from our intervals.

Step 3: Test the intervals

Now, we take a number from each interval and plug it into the inequality to test the sign. From \((-∞, -4)\), for \(x = -5\), the inequality is positive. From \((-4, -2)\), for \(x = -3\), the inequality is negative. From \((-2, 1)\), for \(x = 0\), the inequality is positive. From \((1, ∞)\), for \(x = 2\), the inequality is negative.

Step 4: Provide the solution

Based on the sign test, the solution to the inequality is where the test is negative or zero. Therefore, the solution set to the given inequality will be \([-4, -2) \cup [1, ∞)\] in interval notation.