Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. 2^x2 + 5 ≤ 11x

Key Concepts

Quadratic Inequalities

Quadratic inequalities are expressions that involve a quadratic polynomial set in relation to a value, typically using symbols like ≤, ≥, <, or >. To solve these inequalities, one must first find the roots of the corresponding quadratic equation, which helps determine the intervals to test for the inequality's truth. The solution set is then expressed in interval notation, indicating the ranges of values that satisfy the inequality.

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 interval) or excluded (open interval). For example, (a, b) represents all numbers between a and b, not including a and b, while [a, b] includes both endpoints. This notation is essential for clearly communicating the solution set of inequalities.

Graphing Quadratic Functions

Graphing quadratic functions involves plotting the parabola represented by the quadratic equation on a coordinate plane. The shape of the graph is determined by the coefficients of the quadratic terms, and the vertex represents the maximum or minimum point. Understanding the graph helps visualize the solution to the inequality, as it shows where the function is above or below a certain value, aiding in identifying the intervals that satisfy the inequality.