Problem 48 - College Algebra

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 inequality is \( x \in (-\infty, -3] \cup (-2, \infty) \).

Step 1: Rearrange the inequality

Write the inequality as \( \frac{-x-3}{x+2} \leq 0 \). Here, it is already in that form.

Step 2: Find the critical points

Critical points occur where the numerator or denominator equals to zero. Set \( -x-3 = 0 \) and \( x+2 = 0 \) to find values of \( x \). Solve for \( x \) in each equation. This gives \( x = -3 \) and \( x = -2 \).

Step 3: Test the intervals

Now we have 3 intervals divided by the critical points: \( (-\infty, -3) \), \( (-3, -2) \), and \( (-2, \infty) \). Choose any number within these intervals and put it in the rearranged equation. If the equation is true, include the interval in the solution set. When \( x < -3 \), say \( x = -4 \), the expression becomes negative. When \( -3 < x < -2 \), say \( x = -2.5 \), the expression becomes positive. When \( x > -2 \), say \( x = 0 \), the expression is negative again. Since we are finding where the inequality is less than or equals to zero, we take the intervals where the expression is negative or zero.

Key Concepts

Critical Points

Critical points are essential for solving rational inequalities, as they help divide the number line into manageable intervals. These points occur where the rational expression's numerator or denominator equals zero. To find the critical points, set the numerator and denominator each to zero separately and solve for the variable.
In this exercise, the numerator is \( -x - 3 \) and the denominator is \( x + 2 \). Solving \( -x - 3 = 0 \) gives \( x = -3 \), and solving \( x + 2 = 0 \) gives \( x = -2 \). These solutions, \( x = -3 \) and \( x = -2 \), are the critical points.
Critical points help to identify intervals of the number line to test the inequality, helping determine where it holds true. Remember, the sign of the rational expression can change at these points, so it's crucial to test intervals around them.

Interval Notation

Interval notation is a system used to describe sets of numbers—typically solutions to inequalities—clearly and concisely. It uses parentheses or brackets to indicate whether endpoints are included.

• Use square brackets \( [ ] \) if the endpoint is included (known as "closed"), which corresponds to \