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≥9x^2

Verified step by step guidance

1. Step 1: Start by setting the inequality to zero: $x^3-9x^2\ge 0$.
2. Step 2: Factor the polynomial on the left side: $x^2(x-9)\ge 0$.
3. Step 3: Identify the critical points by setting each factor equal to zero: $x^2=0$ and $x-9=0$. This gives the critical points $x=0$ and $x=9$.
4. Step 4: Use the critical points to divide the number line into intervals: $(-\mathrm{\infty },0)$, $(0,9)$, and $(9,\mathrm{\infty })$. Test a point from each interval in the inequality to determine where the inequality holds.
5. Step 5: Based on the test results, determine the intervals where the inequality is true and express the solution set in interval notation. Remember to include the critical points where the inequality is $\ge$.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to a value using inequality symbols (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.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, [a, b] includes both a and b, while (a, b) does not include them.

Graphing Solutions on a Number Line

Graphing solutions on a number line visually represents the solution set of an inequality. Each interval is marked according to whether it is included or excluded, helping to illustrate the values that satisfy the inequality. This graphical representation aids in understanding the solution's context and range.