Solve each rational inequality. Give the solution set in interval notation.(2x-3)(3x+8)/(x-6)^3≥0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero: \((2x-3)(3x+8) = 0\) and \((x-6)^3 = 0\).
2. Solve \((2x-3)(3x+8) = 0\) to find the zeros of the numerator: \(2x-3=0\) gives \(x=\frac{3}{2}\) and \(3x+8=0\) gives \(x=-\frac{8}{3}\).
3. Solve \((x-6)^3 = 0\) to find the zero of the denominator: \(x=6\).
4. Plot these critical points on a number line: \(-\frac{8}{3}, \frac{3}{2}, 6\).
5. Test intervals between the critical points to determine where the inequality \(\frac{(2x-3)(3x+8)}{(x-6)^3} \geq 0\) holds true, and express the solution in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set against a comparison, such as greater than or equal to zero. To solve these inequalities, one must determine where the rational expression is positive or negative, which often requires finding critical points where the numerator or denominator equals zero.

Critical Points and Test Intervals

Critical points are values of the variable that make the numerator or denominator zero, leading to potential sign changes in the rational expression. After identifying these points, the number line is divided into intervals, and test points from each interval are used to determine whether the expression is positive or negative within those 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 accurately.