Solve each inequality. Give the solution set in interval notation. (2x-1)(x+5)<0

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Identify the critical points by setting each factor equal to zero: solve \(2x - 1 = 0\) and \(x + 5 = 0\) to find the values of \(x\) where the expression changes sign.

Solve \(2x - 1 = 0\) to get \(x = \frac{1}{2}\), and solve \(x + 5 = 0\) to get \(x = -5\). These points divide the number line into three intervals: \((-\infty, -5)\), \((-5, \frac{1}{2})\), and \((\frac{1}{2}, \infty)\).

Choose a test point from each interval and substitute it into the inequality \((2x - 1)(x + 5) < 0\) to determine if the product is positive or negative in that interval.

Based on the sign of the product in each interval, identify which intervals satisfy the inequality \((2x - 1)(x + 5) < 0\) (where the product is negative).

Write the solution set as the union of intervals where the inequality holds true, using interval notation.

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주요 개념

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

Factoring and Zero-Product Property

Understanding how to factor expressions and identify zeros is essential. The inequality involves a product of two linear factors, so finding where each factor equals zero helps determine critical points that divide the number line into intervals for testing.

Sign Analysis on Intervals

After finding critical points, the number line is split into intervals. Testing the sign of the product in each interval determines where the inequality holds true. This method helps identify which intervals satisfy the inequality (less than zero).

Solve each inequality. Give the solution set in interval notation. (2x-1)(x+5)<0

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Factoring and Zero-Product Property

Sign Analysis on Intervals

Interval Notation