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+2

Verified step by step guidance

1. Rewrite the inequality so that all terms are on one side, resulting in a standard polynomial inequality form: \(x^2 - 2x - 2 \leq 0\).
2. Find the roots of the quadratic equation \(x^2 - 2x - 2 = 0\) by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=-2\), and \(c=-2\).
3. Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots and then find the exact roots.
4. Use the roots to divide the real number line into intervals. Test a point from each interval in the inequality \(x^2 - 2x - 2 \leq 0\) to determine where the inequality holds true.
5. Express the solution set as an interval or union of intervals based on the test results, and then 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 (e.g., ≤, <, >, ≥). Solving them requires finding all values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Factoring and Solving Quadratic Equations

To solve polynomial inequalities, especially quadratics, it is essential to rewrite the inequality in standard form and factor the polynomial if possible. Factoring helps identify critical points (roots) where the expression equals zero, which divide the number line into intervals for testing.

Interval Notation and Graphing Solution Sets

After determining the intervals where the inequality holds, solutions are expressed using interval notation, which concisely represents all values satisfying the inequality. Graphing these intervals on a real number line visually shows the solution set, indicating included or excluded endpoints.