Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. x^2 + x - 30 ≤ 0

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 find the values of the variable that satisfy the inequality, often by determining the roots of the corresponding quadratic equation and analyzing the sign of the quadratic expression in the intervals defined by these roots.

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, the interval [a, b] includes both a and b, while (a, b) does not include them. This notation is essential for expressing the solution set of inequalities succinctly.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of its linear factors. This process is crucial for solving quadratic inequalities, as it allows one to identify the roots of the equation, which are the points where the expression equals zero. For example, the quadratic x^2 + x - 30 can be factored into (x - 5)(x + 6), making it easier to analyze the sign of the expression across different intervals.