Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. (3x + 7)/(x - 3) ≤ 0

Verified step by step guidance

1. Identify the critical points by setting the numerator and the denominator equal to zero: \(3x + 7 = 0\) and \(x - 3 = 0\).
2. Solve the equations from step 1 to find the critical points: \(x = -\frac{7}{3}\) and \(x = 3\).
3. Use the critical points to divide the number line into intervals: \((-\infty, -\frac{7}{3})\), \((-\frac{7}{3}, 3)\), and \((3, \infty)\).
4. Test a point from each interval in the inequality \(\frac{3x + 7}{x - 3} \leq 0\) 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 whether the endpoints satisfy the inequality.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in an inequality form. To solve them, one must determine where the rational expression is positive, negative, or zero. This often requires finding critical points where the numerator or denominator equals 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. 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 solution sets of inequalities clearly.

Sign Analysis

Sign analysis is a method used to determine the sign (positive or negative) of a rational expression over different intervals. After identifying critical points, one tests values from each interval to see if the expression is greater than or less than zero. This process is crucial for solving inequalities, as it helps to establish where the inequality holds true.