Solve each rational inequality. Give the solution set in interval notation. (2x-3)/(x²+1)≥0

Verified step by step guidance

1. Identify the rational inequality given: \(\frac{2x - 3}{x^2 + 1} \geq 0\).
2. Note that the denominator \(x^2 + 1\) is always positive for all real \(x\) because \(x^2 \geq 0\) and adding 1 makes it strictly positive. Therefore, the sign of the expression depends only on the numerator \$2x - 3$.
3. Set the numerator greater than or equal to zero to find critical points: \(2x - 3 \geq 0\).
4. Solve the inequality for \(x\): add 3 to both sides and then divide by 2, giving \(x \geq \frac{3}{2}\).
5. Since the denominator is always positive, the solution to the inequality is all \(x\) such that \(x \geq \frac{3}{2}\). Express this solution in interval notation as \(\left[ \frac{3}{2}, \infty \right)\).

Key Concepts

Rational Inequalities

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

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 values from 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 (open intervals) and brackets for values included (closed intervals). Understanding this notation helps clearly communicate the range of solutions.