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 + x ≥ 0

Verified step by step guidance

1. Rewrite the inequality to standard form: \(-x^{2} + x \geq 0\).
2. Factor the left-hand side expression: factor out the common term \(x\), so it becomes \(x(-x + 1) \geq 0\).
3. Identify the critical points by setting each factor equal to zero: \(x = 0\) and \(-x + 1 = 0\) which simplifies to \(x = 1\).
4. Determine the sign of the product \(x(-x + 1)\) on the intervals determined by the critical points: \((-\infty, 0)\), \((0, 1)\), and \((1, \infty)\).
5. Use the sign analysis to find where the product is greater than or equal to zero, then express the solution set in interval notation and graph it on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or 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 Quadratic Expressions

Factoring is the process of rewriting a quadratic polynomial as a product of simpler polynomials. This helps identify the roots or zeros of the polynomial, which are critical points where the expression changes sign, aiding in solving inequalities.

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 inequality holds true, using open or closed dots to indicate whether endpoints are included.