Solve each quadratic inequality. Give the solution set in interval notation. x(x+1)<12

검증된 단계별 안내

Rewrite the inequality in standard form by moving all terms to one side: \(x(x+1) - 12 < 0\).

Expand the left side: \(x^2 + x - 12 < 0\).

Factor the quadratic expression: find two numbers that multiply to \(-12\) and add to \(1\), then write \(x^2 + x - 12\) as \((x + 4)(x - 3)\).

Set each factor equal to zero to find critical points: \(x + 4 = 0\) gives \(x = -4\), and \(x - 3 = 0\) gives \(x = 3\).

Use the critical points to divide the number line into intervals \((-\infty, -4)\), \((-4, 3)\), and \((3, \infty)\), then test a value from each interval in the inequality \((x + 4)(x - 3) < 0\) to determine where the product is negative.

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.

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value. Solving it requires finding the range of x-values that satisfy the inequality, often by analyzing the related quadratic equation and its graph.

Factoring and Solving Quadratic Equations

To solve quadratic inequalities, first rewrite the inequality in standard form and solve the corresponding quadratic equation by factoring or using the quadratic formula. The roots divide the number line into intervals to test for inequality solutions.

Interval Notation and Test Intervals

After finding critical points from the quadratic equation, use interval notation to express solution sets. Test values from each interval determine where the inequality holds true, allowing you to write the solution as a union of intervals.