Solve each rational inequality. Give the solution set in interval notation.(5x-3)^3/(25-8x)^2≤0

Verified step by step guidance

1. Identify the critical points by setting the numerator \((5x-3)^3\) equal to zero and solving for \(x\).
2. Identify the critical points by setting the denominator \((25-8x)^2\) equal to zero and solving for \(x\).
3. Determine the intervals on the number line using the critical points found in the previous steps.
4. Test each interval by selecting a test point and substituting it into the inequality to determine if the inequality holds in that interval.
5. Combine the intervals where the inequality holds true, considering the nature of the inequality (\(\leq 0\)) to include or exclude endpoints, and express the solution set in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set against a constant, often requiring the determination of where the expression is less than, greater than, or equal to zero. To solve these inequalities, one must find the critical points where the expression equals zero or is undefined, and then test intervals to determine where the inequality holds true.

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 greater than 'a' and up to and including 'b'.

Critical Points

Critical points are values of the variable where the rational expression is either zero or undefined. These points are essential in solving rational inequalities as they divide the number line into intervals. By testing these intervals, one can determine where the inequality is satisfied, leading to the final solution set.