Solve each quadratic inequality. Give the solution set in interval notation. (x-4)(x + √2) < 0

Verified step by step guidance

1. Start by identifying the critical points of the inequality by setting each factor equal to zero: solve \(x - 4 = 0\) and \(x + \sqrt{2} = 0\).
2. The solutions to these equations are the critical points \(x = 4\) and \(x = -\sqrt{2}\). These points divide the number line into three intervals: \(( -\infty, -\sqrt{2} )\), \(( -\sqrt{2}, 4 )\), and \(( 4, \infty )\).
3. Choose a test point from each interval and substitute it into the expression \((x - 4)(x + \sqrt{2})\) to determine the sign (positive or negative) of the product in that interval.
4. Since the inequality is \((x - 4)(x + \sqrt{2}) < 0\), select the intervals where the product is negative based on your test points.
5. Express the solution set as the union of intervals where the product is negative, using interval notation and excluding the critical points since the inequality is strict (less than zero, not less than or equal to zero).

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set to be greater than or less than zero. Solving it requires finding the values of the variable that make the inequality true, often by analyzing the sign of the quadratic expression over different intervals.

Factoring Quadratic Expressions

Factoring breaks down a quadratic expression into the product of two binomials. This helps identify the roots or zeros of the quadratic, which are critical points that divide the number line into intervals for testing the inequality.

Interval Notation and Sign Analysis

Interval notation expresses the solution set as ranges of values. After finding the roots, sign analysis determines where the product of factors is positive or negative by testing points in each interval, allowing the correct intervals to be selected for the inequality.