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 separately: solve \((5 - 3x)^2 = 0\) and \((2x - 5)^3 = 0\).
2. Solve \((5 - 3x)^2 = 0\) to find the value(s) of \(x\) where the numerator is zero, and solve \((2x - 5)^3 = 0\) to find the value(s) of \(x\) where the denominator is zero.
3. Use the critical points to divide the real number line into intervals. These intervals will be tested to determine where the inequality holds.
4. Choose a test point from each interval and substitute it into the expression \(\frac{(5 - 3x)^2}{(2x - 5)^3}\) to check whether the expression is positive or negative in that interval.
5. Based on the sign of the expression in each interval and the inequality \(> 0\), write the solution set in interval notation, excluding any points where the denominator is zero (since the expression is undefined there).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares the quotient to zero or another value. Solving them requires finding where the expression is positive, negative, or zero by analyzing the signs of numerator and denominator.

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator zero, dividing the number line into intervals. By testing points in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation

Interval notation is a concise way to express solution sets using parentheses and brackets to indicate open or closed intervals. It clearly shows the range of values satisfying the inequality, excluding points where the expression is undefined or does not meet the inequality.