Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2>16

Verified step by step guidance

1. Rewrite the inequality as a quadratic equation: \(x^2 = 16\).
2. Solve the quadratic equation \(x^2 = 16\) by taking the square root of both sides, which gives \(x = 4\) and \(x = -4\).
3. These solutions divide the number line into three intervals: \((-\infty, -4)\), \((-4, 4)\), and \((4, \infty)\).
4. Test a point from each interval in the original inequality \(x^2 > 16\) to determine which intervals satisfy the inequality.
5. Based on the test results, write the solution set in interval notation, including only the intervals where the inequality holds true.

Key Concepts

Quadratic Inequalities

Quadratic inequalities are expressions that involve a quadratic polynomial set in relation to a value, typically using inequality symbols such as '>', '<', '≥', or '≤'. To solve these inequalities, one must determine the values of the variable that satisfy the inequality, often by finding the roots of the corresponding quadratic equation and testing intervals.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to and including 5.

Graphing Quadratics

Graphing quadratics involves plotting the quadratic function on a coordinate plane to visualize its shape, which is a parabola. The vertex, axis of symmetry, and intercepts are key features that help in understanding the behavior of the quadratic function, particularly in determining where it is above or below a certain value, which is essential for solving inequalities.