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 + 2x ≥ 0

Verified step by step guidance

1. Rewrite the inequality to a standard form: \(-x^2 + 2x \geq 0\).
2. Factor the left-hand side expression. First, factor out the negative sign: \(-(x^2 - 2x) \geq 0\). Then factor the quadratic inside the parentheses: \(-(x(x - 2)) \geq 0\).
3. Multiply both sides of the inequality by \(-1\) to remove the negative sign, remembering to reverse the inequality sign because you are multiplying by a negative number: \(x(x - 2) \leq 0\).
4. Find the critical points by setting each factor equal to zero: \(x = 0\) and \(x - 2 = 0 \Rightarrow x = 2\). These points divide the number line into intervals to test.
5. Test values from each interval (\((-\infty, 0)\), \((0, 2)\), and \((2, \infty)\)) in the inequality \(x(x - 2) \leq 0\) to determine where the inequality holds true, 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 quadratic, 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.