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Chapter 2: Problem 88

Solve each inequality using a graphing utility. $$x^{3}+x^{2}-4 x-4>0$$

Short Answer

Expert verified
The solution to the inequality will be the x-values for which the function plot is above the x-axis. Since the graphing utility is required, the exact solution intervals aren't known without it.

Step by step solution

01

Plot the Cubic Function

First, construct the graph of \(y = x^{3}+x^{2}-4x-4\) using a graphing utility. Note the x-intercepts and how the plot moves across the x-axis.
02

Identify the Solution Intervals

Identify the intervals on the x-axis where the function \(y = x^{3}+x^{2}-4x-4\) is greater than 0. This means looking for sections where the graph is above the x-axis.
03

Write Down the Solution

Write down the solution as an interval or intervals on the x-axis where the graph is above the x-axis. The solution isn't a specific value, but rather ranges of x-values for which the original inequality is true. If there are multiple intervals, they should be separated by the union symbol 鈭� to represent 'or'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cubic Functions
Cubic functions are polynomial functions of degree three. They take the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a eq 0 \). These functions can have various shapes and behaviors due to their higher-degree terms. When graphing cubic functions, you often look for features like turning points or intercepts, especially the x-intercepts. These points indicate where the graph crosses the x-axis and potentially changes direction. The number of real x-intercepts can vary鈥攖here can be one, two, or three real x-intercepts. Each of these intercepts represents a potential root of the function.
Graphing Inequalities
Graphing inequalities involves plotting the related equation and identifying the areas where the inequality holds. For our exercise, the inequality is \( x^3 + x^2 - 4x - 4 > 0 \). This means we're interested in the parts of the graph where this function is above the x-axis. To do this:
  • First, plot the equation \( y = x^3 + x^2 - 4x - 4 \) on a graph.
  • Check where the line is positioned relative to the x-axis.
  • Identify intervals where the curve lies above the x-axis because this indicates where the function values are positive.
These are the intervals where the inequality \( x^3 + x^2 - 4x - 4 > 0 \) holds true. By using graphing utilities, it becomes easier to visualize these areas, especially for more complex functions like cubics.
Solution Intervals
Solution intervals are ranges on the x-axis that satisfy the inequality condition. In the context of our cubic function \( x^3 + x^2 - 4x - 4 > 0 \), solution intervals are the parts of the x-axis where the function is positive. To find these intervals using a graph:
  • Look for the x-intercepts; these are potential changes where the function may shift from negative to positive values, or vice versa.
  • The intervals between these intercepts need exploration to determine whether the function remains above or below the x-axis.
  • The positive intervals where the graph stays above the x-axis are written as \((a, b)\), indicating the range from \(a\) to \(b\).
If there are multiple positive intervals, they should be combined using the union symbol \( \cup \), indicating that any of these intervals satisfy the inequality. Understanding solution intervals helps in pinpointing specific ranges of x-values that make the inequality true.

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Most popular questions from this chapter

Write the equation of each parabola in standard form. Vertex: \((-3,-4) ;\) The graph passes through the point \((1,4)\)

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Does the equation \(3 x+y^{2}=10\) define \(y\) as a function of \(x ?\) (Section \(1.2,\) Example 3 )

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