Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. - ( x +√2)(x-3) < 0

Verified step by step guidance

1. First, identify the critical points by setting each factor equal to zero: solve \( x + \sqrt{2} = 0 \) and \( x - 3 = 0 \). These give the points \( x = -\sqrt{2} \) and \( x = 3 \).
2. Next, use these critical points to divide the number line into three intervals: \( (-\infty, -\sqrt{2}) \), \( (-\sqrt{2}, 3) \), and \( (3, \infty) \).
3. Choose a test point from each interval and substitute it into the inequality \( -(x + \sqrt{2})(x - 3) < 0 \) to determine whether the inequality holds in that interval.
4. Analyze the sign of each factor and the overall expression at the test points to decide which intervals satisfy the inequality.
5. Finally, write the solution set as the union of intervals where the inequality is true, using interval notation.

Key Concepts

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 the factors helps identify the critical points where the expression changes sign.

Sign Analysis of Inequalities

To solve inequalities like (x + √2)(x - 3) < 0, determine where the product is negative. This involves analyzing the sign of each factor in intervals defined by the roots (x = -√2 and x = 3) and finding where their product is less than zero.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities. It uses parentheses and brackets to indicate open or closed intervals. For this inequality, the solution set will be expressed as intervals where the product is negative.