Solve each rational inequality. Give the solution set in interval notation.

Verified step by step guidance

1. Identify the rational inequality to solve: \(\frac{9x - 8}{4x^2 + 25} < 0\).
2. Determine the critical points by setting the numerator and denominator equal to zero separately: solve \$9x - 8 = 0\( and \)4x^2 + 25 = 0$.
3. Note that \$4x^2 + 25 = 0\( has no real solutions because \)4x^2 = -25\( is impossible for real \)x$, so the denominator is always positive.
4. Since the denominator is always positive, the sign of the rational expression depends solely on the numerator \$9x - 8\(; find where \)9x - 8 < 0$.
5. Solve \$9x - 8 < 0$ to find the solution set, then express the solution in interval notation.

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 understanding where the expression is positive, negative, or zero by analyzing the signs of numerator and denominator.

Sign Analysis and Critical Points

To solve rational inequalities, identify critical points where the numerator or denominator equals zero. These points divide the number line into intervals. Testing each interval determines where the inequality holds true, considering that division by zero is undefined.

Interval Notation

Interval notation is a concise way to express solution sets of inequalities. It uses parentheses for values not included (like points causing zero denominators) and brackets for included endpoints. Understanding this notation helps clearly communicate the solution.