Solve each quadratic inequality. Give the solution set in interval notation. - ( x +√2)(x-3) < 0

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Start by identifying the critical points of the inequality by setting each factor equal to zero: solve \( x + \sqrt{2} = 0 \) and \( x - 3 = 0 \).

Find the roots from these equations: \( x = -\sqrt{2} \) and \( x = 3 \). These points divide the number line into three intervals to test.

Determine the sign of the product \( (x + \sqrt{2})(x - 3) \) in each interval: \( (-\infty, -\sqrt{2}) \), \( (-\sqrt{2}, 3) \), and \( (3, \infty) \).

Choose a test point from each interval and substitute it into the expression \( (x + \sqrt{2})(x - 3) \) to check if the product is less than zero in that interval.

Based on the sign test, write the solution set as the union of intervals where the product is negative, using interval notation.

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주요 개념

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

Factoring Quadratic Expressions

Factoring involves expressing a quadratic expression as a product of two binomials. In this problem, the quadratic is already factored as (x + √2)(x - 3). Recognizing this form helps identify the roots or zeros of the quadratic, which are critical for solving inequalities.

Solving Quadratic Inequalities

Solving quadratic inequalities requires determining where the quadratic expression is less than, greater than, or equal to zero. This involves analyzing the sign of the product in intervals defined by the roots. Testing points in each interval helps find where the inequality holds true.

Interval Notation

Interval notation is a way to represent sets of numbers between two endpoints. It uses parentheses for open intervals (excluding endpoints) and brackets for closed intervals (including endpoints). Expressing the solution set in interval notation clearly communicates the range of values satisfying the inequality.