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

Verified step by step guidance

1. Identify the inequality: $\frac{2x-3}{x^2+1}\ge 0$.
2. Recognize that the denominator $x^2+1$ is always positive since it is a sum of squares.
3. Focus on the numerator $2x-3$ to determine when the expression is non-negative.
4. Solve the equation $2x-3=0$ to find the critical point where the expression changes sign.
5. Determine the intervals on the number line and test points in each interval to see where the inequality holds true.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to an inequality (e.g., ≥, ≤, >, <). To solve these, one must determine where the rational expression is positive or negative, which often requires finding critical points where the numerator or denominator equals zero.

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 interval) or excluded (open interval). For example, (a, b) means all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Critical Points and Test Intervals

Critical points are values of the variable that make the rational expression equal to zero or undefined. To solve the inequality, one identifies these points and tests intervals between them to determine where the expression satisfies the inequality. This process helps in constructing the solution set in interval notation.