College Algebra Chapter 3 Polynomial and Rational Functions Section 3.6 Polynomial and Rational Inequalities

Concepts 1. Solve Polynomial Inequalities 2. Solve Rational Inequalities 3. Solve Applications Involving Polynomial and Rational Inequalities

Solve Polynomial Inequalities To Solve a Nonlinear Inequality f (x) < 0, f (x) > 0, f (x) ≤ 0, or f (x) ≥ 0: Step 1: Rearrange the terms of the inequality so that one side is set to 0.   Step 2: Find the real solutions of the related equation f (x) = 0 and any values of x that make f (x) undefined. These are the “boundary” points for the solution set to the inequality.

Solve Polynomial Inequalities Step 3: Determine the sign of f (x) on the intervals defined by the boundary points. If f (x) is positive, then the values of x on the interval are solutions to f (x) > 0. If f (x) is negative, then the values of x on the interval are solutions to f (x) < 0.   Step 4: Determine whether the boundary points are included in the solution set. Step 5: Write the solution set.

Example 1: Solve for x.

Example 2: Solve for x.

Example 3: Solve for x.

Example 4: Solve for x.

Example 5: Solve for x.

Concepts 1. Solve Polynomial Inequalities 2. Solve Rational Inequalities 3. Solve Applications Involving Polynomial and Rational Inequalities

Example 6: Solve for x.

Example 7: Solve for x.

Example 7 continued:

Example 8: Solve for x.

Example 8 continued:

Concepts 1. Solve Polynomial Inequalities 2. Solve Rational Inequalities 3. Solve Applications Involving Polynomial and Rational Inequalities

Example 9: Fumio is constructing a circular koi pond in his back yard. What range of values may he have for the diameter of the pond if the area is to be no more than 78.5 square feet? What is the largest possible diameter? (Use the approximation )

Example 9 continued:

Example 10: The cost C (in dollars) to produce x items is given by the function below. If the cost is to be kept below $20, how many items may be produced?

Example 10 continued: