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

검증된 단계별 안내

Rewrite the inequality so that all terms are on one side, resulting in a standard polynomial inequality form: \(x^2 - 2x - 2 \leq 0\).

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 \(c=-2\).

Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots and then find the exact roots.

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.

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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