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. $x^3\ge 9x^2$

Verified step by step guidance

1. Start by rewriting the inequality \(x^3 \geq 9x^2\) so that one side is zero: subtract \$9x^2$ from both sides to get \(x^3 - 9x^2 \geq 0\).
2. Factor the left-hand side expression: factor out the greatest common factor \(x^2\), giving \(x^2(x - 9) \geq 0\).
3. Identify the critical points by setting each factor equal to zero: \(x^2 = 0\) gives \(x = 0\), and \(x - 9 = 0\) gives \(x = 9\). These points divide the number line into intervals to test.
4. Test the sign of the expression \(x^2(x - 9)\) in each interval determined by the critical points: \((-\infty, 0)\), \((0, 9)\), and \((9, \infty)\). Remember that \(x^2\) is always nonnegative, so the sign depends mainly on \((x - 9)\).
5. Based on the sign analysis, determine where the inequality \(x^2(x - 9) \geq 0\) holds true, include points where the expression equals zero, and express the solution set in interval notation. Finally, graph this solution set on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to another value using inequality symbols (>, ≥, <, ≤). 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 Polynomials

Factoring is the process of rewriting a polynomial as a product of simpler polynomials or factors. It helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign in inequality problems.

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 solution lies, using open or closed dots to indicate whether endpoints are included or excluded.