Solve each rational inequality. Give the solution set in interval notation. (9x-11)(2x+7)/(3x-8)^3>0

Verified step by step guidance

1. Identify the critical points by setting each factor in the inequality equal to zero: $9x-11=0$, $2x+7=0$, and $(3x-8{)}^3=0$.
2. Solve each equation to find the critical points: $x=\frac{11}{9}$, $x=-\frac{7}{2}$, and $x=\frac{8}{3}$.
3. Use these critical points to divide the number line into intervals: $(-\mathrm{\infty },-\frac{7}{2})$, $(-\frac{7}{2},\frac{11}{9})$, $(\frac{11}{9},\frac{8}{3})$, and $(\frac{8}{3},\mathrm{\infty })$.
4. Test a point from each interval in the inequality $(9x-11)(2x+7)/(3x-8{)}^3>0$ to determine where the inequality holds true.
5. Combine the intervals where the inequality is satisfied to express the solution set in interval notation.

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 or negative. This typically involves finding critical points where the numerator or denominator equals zero, and then testing intervals around these points to see where the inequality holds true.

Critical Points

Critical points are values of the variable that make the numerator or denominator of a rational expression equal to zero. These points are essential for determining the intervals to test in the inequality. In the given inequality, the critical points are found by solving (9x-11)(2x+7) = 0 and (3x-8)^3 = 0, which helps in identifying where the expression changes sign.

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 all numbers between a and b but not a and b themselves, while [a, b] includes a and b. This notation is crucial for expressing the solution set of inequalities clearly.