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. x²−6x+9<0

Verified step by step guidance

1. Recognize that the inequality is a quadratic inequality: \(x^2 - 6x + 9 < 0\).
2. Factor the quadratic expression on the left side. Notice that \(x^2 - 6x + 9\) is a perfect square trinomial, so it factors as \((x - 3)^2\).
3. Rewrite the inequality using the factorization: \((x - 3)^2 < 0\).
4. Consider the properties of squares: since \((x - 3)^2\) is always greater than or equal to zero for all real \(x\), it can never be less than zero.
5. Conclude that there are no real values of \(x\) that satisfy \((x - 3)^2 < 0\), so the solution set is the empty set, which in interval notation is \(\emptyset\).

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols like <, >, ≤, or ≥. Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Factoring Quadratic Expressions

Factoring is the process of rewriting a quadratic expression as a product of two binomials. For example, x² - 6x + 9 factors to (x - 3)(x - 3). Factoring helps identify the roots of the polynomial, which are critical points for determining where the inequality changes sign.

Interval Notation and Number Line Graphing

Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. Graphing on a number line visually shows where the polynomial is positive or negative. Together, they help communicate the solution clearly, indicating which intervals satisfy the inequality.