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. (−x−3)/(x+2)≤0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero: $-x-3=0$ and $x+2=0$.
2. Solve the equations from step 1 to find the critical points: $x=-3$ and $x=-2$.
3. Determine the intervals to test by using the critical points: $(-\mathrm{\infty },-3)$, $(-3,-2)$, and $(-2,\mathrm{\infty })$.
4. Choose a test point from each interval and substitute it into the inequality $\frac{-x-3}{x+2}\le 0$ to determine where the inequality holds true.
5. Combine the intervals where the inequality is satisfied and express the solution set in interval notation, considering the critical points where the inequality is equal to zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in an inequality format. To solve them, one must determine where the rational expression is positive, negative, or zero. This often requires finding critical points by setting the numerator and denominator to zero, which helps in analyzing the sign of the expression across different intervals.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (a, b] includes all numbers greater than 'a' and up to 'b', including 'b' itself.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the solutions to an inequality. This involves marking critical points and shading the regions that satisfy the inequality. Understanding how to accurately depict these regions helps in interpreting the solution set and communicating the results effectively.