Problem 48 - 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. $$ \frac{-x-3}{x+2} \leq 0 $$

Step-by-Step Solution

Answer

The solution to the given inequality is \(-∞, -3] U (-3, -2\).

Step 1: Identify the Critical Points

Critical points are the values of x that make the rational function undefined or change its sign. Begin by identifying those points. Here, the function will be undefined where the denominator is zero, i.e when x = -2. The sign will change where the numerator is zero, i.e when x = -3.

Step 2: Build the Number Line and Identify the Intervals

Draw a number line and plot the identified critical points. This divides the number line into three intervals: (-∞, -3), (-3, -2) and (-2, ∞).

Step 3: Test the Intervals

Choose test points from each interval and substitute them into the original inequality to determine if the resulting value is greater than, equal to, or less than zero. Let -4 be the test point for (-∞, -3), -2.5 for (-3, -2), and 0 for (-2, ∞). After substitution, for -4 we get positive value, for -2.5 positive value and for 0 we get negative value. Thus, intervals (-∞, -3) and (-3, -2) are in the solution set.

Step 4: Write the Solution in Interval Notation and Graph

The solution set is x in (-∞, -3] U (-3, -2). Plot this solution set on the real number line. A closed dot on -3 indicates that -3 is included and an open dot on -2 indicats that -2 is excluded.