Solve each rational inequality. Give the solution set in interval notation. (5-3x)^2/(2x-5)^3>0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero: $(5-3x{)}^2=0$ and $(2x-5{)}^3=0$.
2. Solve $(5-3x{)}^2=0$ to find the critical point for the numerator. This gives $5-3x=0$, leading to $x=\frac{5}{3}$.
3. Solve $(2x-5{)}^3=0$ to find the critical point for the denominator. This gives $2x-5=0$, leading to $x=\frac{5}{2}$.
4. Determine the intervals to test by using the critical points $x=\frac{5}{3}$ and $x=\frac{5}{2}$. The intervals are $(-\mathrm{\infty },\frac{5}{3})$, $(\frac{5}{3},\frac{5}{2})$, and $(\frac{5}{2},\mathrm{\infty })$.
5. Test each interval by selecting a test point from each interval and substituting it into the inequality $(5-3x{)}^2/(2x-5{)}^3>0$ to determine where the inequality holds true. Use the results to write the solution set in interval notation.

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 or negative. This typically involves finding critical points where the numerator or denominator equals zero and testing intervals around these points to establish the sign of the expression.

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, (a, b) represents all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Critical Points

Critical points are values of the variable where the rational expression is either zero or undefined. These points are essential for determining the intervals to test in the inequality. In the given inequality, critical points arise from setting the numerator and denominator to zero, which helps in analyzing the sign of the expression across different intervals.