Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. -x^2 + 2x + 6 > 0

Key Concepts

Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not equal. They can be represented using symbols such as '>', '<', '≥', and '≤'. Solving inequalities involves finding the values of the variable that make the inequality true, which often requires similar techniques to solving equations, but with additional considerations for the direction of the inequality when multiplying or dividing by negative numbers.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the properties of quadratic functions, such as their vertex, axis of symmetry, and intercepts, is crucial for analyzing inequalities involving quadratics.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the number line. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (a, b) includes all numbers between a and b but not a and b themselves, while [a, b] includes a and b. Writing solutions to inequalities in interval notation provides a clear and concise way to express the set of values that satisfy the inequality.