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+4)(x−1)/(x+2)≤0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero: \((x+4)(x-1) = 0\) and \(x+2 = 0\).
2. Solve the equations to find the critical points: \(x = -4\), \(x = 1\), and \(x = -2\).
3. Use these critical points to divide the number line into intervals: \((-\infty, -4)\), \((-4, -2)\), \((-2, 1)\), and \((1, \infty)\).
4. Test a point from each interval in the inequality \((x+4)(x-1)/(x+2) \leq 0\) 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 inclusion of endpoints where the inequality is \(\leq\).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to zero. To solve them, one must determine where the rational expression is positive, negative, or zero. This typically involves 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 'a' but not 'b', which is essential for expressing the solution set of inequalities accurately.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the values that satisfy the inequality. This involves marking critical points and shading the appropriate regions based on the sign of the rational expression. Understanding how to graph these solutions helps in quickly identifying the intervals that meet the conditions of the inequality.