33-66 . Solve the nonlinear inequality. Express the solution using interval notation and graph t

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33-66 . Solve the nonlinear inequality. Express the solution...

Key Concepts

Quadratic Inequality

A quadratic inequality is an inequality that involves a quadratic expression, typically written in the form ax^2 + bx + c > 0, < 0, ? 0, or ? 0. Solving such inequalities involves finding the roots of the corresponding quadratic equation and determining the intervals where the quadratic expression meets the inequality condition.

Rewriting the Inequality

To solve a quadratic inequality, the first step is to rewrite the inequality so that one side is zero. This standardization allows you to find the solutions to the associated quadratic equation, which are critical in determining the intervals where the inequality holds.

Finding the Roots / Factoring

The roots of the quadratic equation, obtained by setting the quadratic expression equal to zero, serve as boundary points that partition the number line into intervals. Factoring the quadratic, when possible, or using the quadratic formula are standard techniques to determine these roots.

Sign Analysis

Once the quadratic is factored and the roots are identified, a sign analysis is performed on the intervals defined by these roots. This analysis determines where the quadratic expression is positive or negative, thereby identifying the solution set for the inequality.

Interval Notation

Interval notation is a concise way to represent the set of values that satisfy an inequality. It uses brackets and parentheses to indicate whether endpoints are included or excluded, providing a clear and efficient representation of the solution set.

Graphing the Solution Set

Graphing the solution set involves plotting the intervals on a number line, marking the boundary points, and indicating with open or closed circles whether those points are included in the solution. This visual representation aids in understanding the relationship between the algebraic solution and its geometric interpretation.

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