Solve each polynomial inequality in Exercises 1-42 and graph the solution set on a real number l

step 1 So we're solving this inequality. 3x2 .6 plus 16, x is less than minus 5. And our first step is

Solve each polynomial inequality in Exercises 1-42 and graph...

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set in relation to zero using inequality signs (>, <, ?, or ?). Solving these inequalities typically involves determining where the quadratic expression is positive, negative, or zero. This forms the basis for understanding the behavior of the function and identifying the intervals that satisfy the inequality.

Rearranging the Inequality

Before solving a quadratic inequality, it is important to rearrange it so that all terms are on one side of the inequality sign. This standard form (typically set to zero) allows for easier factorization or the application of the quadratic formula, which simplifies the process of analyzing the function's sign changes.

Finding Zeros or Roots

The zeros or roots of the corresponding quadratic equation (obtained by setting the expression equal to zero) are critical because they divide the number line into intervals. These roots are found by factoring the quadratic expression or using the quadratic formula, and they mark the boundary points where the sign of the quadratic expression may change.

Test Interval Method

After determining the zeros, the number line is segmented into distinct intervals. A test point from each interval is used to determine the sign of the quadratic expression within that interval. This analysis helps identify which segments satisfy the original inequality based on whether the expression is positive or negative.

Interval Notation and Graphing on the Number Line

The final solution is often expressed in interval notation, which provides a concise representation of the set of values that satisfy the inequality. Additionally, graphing the solution on a number line offers a visual representation of the intervals where the inequality holds true, illustrating whether endpoints are included or excluded.

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