Solve each inequality in Exercises 86–91 using a graphing utility. x³ + x² - 4x - 4 > 0

Verified step by step guidance

1. Rewrite the inequality to clearly identify the function: \(x^{3} + x^{2} - 4x - 4 > 0\).
2. Use a graphing utility to graph the function \(f(x) = x^{3} + x^{2} - 4x - 4\) and observe where the graph lies above the x-axis, since the inequality is \(f(x) > 0\).
3. Identify the x-intercepts (roots) of the function from the graph, which are the points where \(f(x) = 0\). These points divide the number line into intervals.
4. Test values from each interval determined by the roots to check whether \(f(x)\) is positive or negative in those intervals.
5. Write the solution as the union of intervals where the function is greater than zero, based on the graph and test values.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols. Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Graphing Utility for Polynomial Functions

A graphing utility is a tool or software that plots the graph of functions, including polynomials. It helps visualize where the polynomial is above or below the x-axis, which corresponds to positive or negative values, aiding in solving inequalities by identifying solution intervals.

Sign Analysis of Polynomial Functions

Sign analysis involves determining where a polynomial function is positive or negative by examining its roots and the behavior between them. After finding the zeros, the sign of the polynomial in each interval is tested to solve inequalities like f(x) > 0.