In Exercises 71–72, use the graph of the polynomial function to solve each inequality.


2x^3 + 11x^2 ≥ 7x + 6

Verified step by step guidance

1. Rewrite the inequality 2x^3 + 11x^2 ≥ 7x + 6 by moving all terms to one side to set the inequality to zero: 2x^3 + 11x^2 - 7x - 6 ≥ 0.
2. Recognize that the inequality now involves the polynomial function f(x) = 2x^3 + 11x^2 - 7x - 6, so the problem reduces to finding where f(x) ≥ 0.
3. Identify the roots of the polynomial from the graph, which are the x-values where the graph crosses the x-axis: x = -1, x = 2/3, and x = 2.
4. Use the roots to divide the number line into intervals: (-∞, -1), (-1, 2/3), (2/3, 2), and (2, ∞).
5. Determine the sign of f(x) on each interval by observing the graph: check whether the graph is above (f(x) > 0) or below (f(x) < 0) the x-axis in each interval, then select the intervals where f(x) ≥ 0 to solve the inequality.

Key Concepts

Polynomial Functions and Their Graphs

A polynomial function is an expression involving variables raised to whole-number exponents and their coefficients. Its graph is a smooth, continuous curve that can cross the x-axis at points called roots or zeros. Understanding the shape and roots of the polynomial helps in analyzing its behavior and solving inequalities.

Finding Roots of a Polynomial

Roots or zeros of a polynomial are the x-values where the function equals zero, i.e., where the graph intersects the x-axis. These points divide the number line into intervals where the polynomial can be positive or negative. Identifying roots is essential for solving inequalities involving polynomials.

Solving Polynomial Inequalities Using Graphs

To solve inequalities like f(x) ≥ 0, analyze the graph to determine where the polynomial is above or on the x-axis. The solution includes intervals where the graph lies on or above zero, including the roots if the inequality is non-strict (≥ or ≤). This graphical approach provides a visual method to solve polynomial inequalities.