Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^2−6x+9<0

Verified step by step guidance

1. Identify the polynomial inequality: $x^2 - 6x + 9 < 0$.
2. Recognize that the expression $x^2 - 6x + 9$ is a perfect square trinomial, which can be rewritten as $(x - 3)^2$.
3. Set the inequality $(x - 3)^2 < 0$ and analyze the expression.
4. Understand that a square of a real number is always non-negative, meaning $(x - 3)^2 \geq 0$ for all real $x$.
5. Conclude that there are no real solutions where $(x - 3)^2 < 0$, hence the solution set is empty.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to a value using inequality signs (e.g., <, >, ≤, ≥). To solve these inequalities, one typically finds the roots of the corresponding polynomial equation and tests intervals between these roots to determine where the inequality holds true.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the intervals where the inequality is satisfied. This involves marking the critical points (roots) and shading the regions that meet the inequality condition, which helps in understanding the solution's range.

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, (a, b) means all numbers between a and b, not including a and b, while [a, b] includes both endpoints.