Problem 88 - College Algebra

Question

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ x^{3}+x^{2}-4 x-4>0 $$

Step-by-Step Solution

Answer

After plotting, you may found two intersections of the function with the x-axis, namely x = -2 and x = 2. By testing with x-values in the three identified regions, it is observed that the inequality holds for two of them. So, the solution will be \(x<-2\) or \(x>2\). But this may be vary according to actual graphing results.

Step 1: Plot the Function

Using your graphing utility, plot \(y = x^{3} + x^{2} - 4x - 4\). This function is a cubic and so it will have a 'w' shape with some areas being above and below the x-axis.

Step 2: Identify Zero Crossing Points

Identify the points where the graph crosses the x-axis. This is because these are the points where \(y = 0\). Since we are finding where \(y > 0\), we need to know these points to identify the range of x values.

Step 3: Identify Sign for each Region

There should be three regions determined by the two zero-crossing points. Use a point in each region (not a zero-crossing point) and evaluate the inequality. If the inequality holds for that point, then it will hold for all points in that region.

Step 4: Solution Set

The solution will be all x-values that make \(y = x^{3} + x^{2} - 4x - 4\) greater than 0. This will be the union of all the valid regions identified in the previous step.

Key Concepts

Cubic Functions

Cubic functions are mathematical expressions where the highest degree of the variable is three, typically written in the form \(y = ax^3 + bx^2 + cx + d\). This type of function has a unique property: it creates a 'w'-shaped curve on a graph, due to the different turning points caused by its cubic nature.

Cubic functions can have up to three real roots and always have at least one real root, which means the graph will intersect the x-axis at least once. Because of this intersection and the shape of the graph, they provide interesting challenges when solving inequalities. Understanding and visualizing these cubics is key to solving equations and inequalities involved with them.

Graphing Utility

Graphing utilities are powerful tools for visualizing mathematical functions such as cubic functions. They help students and learners explore functions to better understand their characteristics without manually plotting many points.

Using a graphing utility, like calculators or software, allows you to:

• See the overall shape of the function.
• Quickly identify the points where the curve crosses the x-axis, known as zero crossing points.
• Gain insights about functions' behavior at various intervals.
• Check for symmetry and turning points.

These utilities save time and can instantly display complex graphs, making analysis and solutions much more accessible.

Zero Crossing Points

Zero crossing points are crucial when solving inequalities, especially for cubic functions. These are the points at which the function value equals zero; in other words, where the graph intersects the x-axis.

Finding these points involves identifying the roots of the equation, so for the function \(y = x^3 + x^2 - 4x - 4\), solving \(x^3 + x^2 - 4x - 4 = 0\) gives us the x-values where the graph crosses the x-axis.

In the inequality \(x^3 + x^2 - 4x - 4 > 0\), these zero crossing points act as boundaries. Between these intersections, the sign of the graph changes, allowing us to determine which parts of the graph are positive or negative. This helps in identifying intervals that satisfy the inequality.

Solution Set

The solution set of an inequality consists of all possible values of the variable that make the inequality true. After using a graphing utility to analyze the function and identify zero crossing points, the next step is to determine which intervals of x-values result in the cubic function being greater than zero.

These intervals are found by checking the sign of the function in regions divided by the zero crossing points. For example, if the function is positive in an interval of x-values between two zero crossing points, then those x-values will be part of the solution set.

The solution set is often expressed as a union of these intervals. For \(x^3 + x^2 - 4x - 4 > 0\), after testing the signs in various regions, you combine the positive intervals to create the complete solution that satisfies the inequality. Understanding this allows students to neatly encapsulate the solution to problems involving cubic functions.