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 rational inequality to solve: \(\frac{(5x-3)^3}{(25-8x)^2} \leq 0\).
2. Determine the critical points by setting the numerator and denominator equal to zero separately: solve \$5x - 3 = 0\( and \)25 - 8x = 0$.
3. Analyze the sign of the numerator and denominator on intervals determined by the critical points. Remember that the denominator squared, \((25 - 8x)^2\), is always non-negative and zero only at its root.
4. Since the denominator cannot be zero (division by zero is undefined), exclude that point from the solution set. Then, find where the entire expression is less than or equal to zero by considering the sign of the numerator and denominator on each interval.
5. Express the solution set in interval notation, including points where the expression equals zero (numerator zero) but excluding points where the denominator is zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio 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 and Domain Restrictions

Interval notation expresses solution sets compactly using parentheses and brackets to indicate open or closed intervals. Domain restrictions exclude values that make the denominator zero, ensuring the solution set only includes valid inputs for the inequality.