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


2x³ + 11x² < 7x + 6

Verified step by step guidance

1. Rewrite the inequality 2x^3 + 11x^2 < 7x + 6 by bringing all terms to one side to set the inequality to zero: $\mathrm{2x^3}+\mathrm{11x^2}-\mathrm{7x}-6<0$.
2. Define a function $f(x)\; =\; 2x^3\; +\; 11x^2\; -\; 7x\; -\; 6$ and analyze where this function is less than zero by using the graph provided.
3. Identify the roots of the polynomial from the graph, which are the x-values where the graph crosses the x-axis. These roots are approximately $x\; =\; -1$, $x\; =\; \backslash frac\{2\}\{3\}$, and $x\; =\; 2$.
4. Determine the intervals on the x-axis divided by these roots: $(-\backslash infty,\; -1)$, $(-1,\; \backslash frac\{2\}\{3\})$, $(\backslash frac\{2\}\{3\},\; 2)$, and $(2,\; \backslash infty)$.
5. Use the graph to check the sign of $f(x)$ on each interval. The solution to the inequality will be the intervals where the graph lies below the x-axis.

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.

Roots (Zeros) of a Polynomial

Roots 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 polynomial inequalities, analyze the graph to determine where the function is above or below the x-axis. For example, to solve f(x) < 0, find intervals where the graph lies below the x-axis. The roots mark boundary points for these intervals, guiding the solution set.