Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. {x^2 - 3x- 4} /{ x^2 + 6x + 9} ≤ 0

Verified step by step guidance

1. 1. Factor the numerator and the denominator of the rational expression.
2. 2. Identify the critical points by setting each factor equal to zero.
3. 3. Use the critical points to divide the number line into intervals.
4. 4. Test a point from each interval in the inequality to determine where the inequality holds true.
5. 5. Write the solution set in interval notation, including endpoints where the inequality is less than or equal to zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to zero, typically using symbols like ≤, ≥, <, or >. To solve these inequalities, one must determine where the rational expression is positive or negative, which often requires finding critical points where the numerator and denominator equal zero.

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 (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2.

Critical Points and Test Intervals

Critical points are the values of the variable that make the numerator or denominator of a rational expression zero. These points divide the number line into intervals, which can be tested to determine the sign of the rational expression in each interval. This process helps identify where the inequality holds true.