Problem 51 In order to solve an inequality ... [FREE SOLUTION]

Chapter 1: Problem 51

In order to solve an inequality on a graphics calculator, we can graph a corresponding function and determine where it is positive and where it is negative. In Exercises $51-52$ use the zoom feature of a graphics calculator to find an approximate solution of the inequality. Zoom until successive values of the $x$ coordinate have identical first three digits. $$ \begin{aligned} &x^{3}+1 \geq-x-2 \text { (Hint: Let } f(x)=x^{3}+x+3 \text { , and determine } \\ &\text { where } f(x) \geq 0 . \text { ) } \end{aligned} $$

Short Answer

Expert verified

The approximate solution for the inequality is where $ x $ is around -2.25 and $ x > -1 $.

Step by step solution

Define the Function

We begin by noting from the hint given in the problem that the function $ f(x) $ is defined as $ f(x) = x^3 + x + 3 $. This function is derived from rearranging the original inequality $ x^3 + 1 \geq -x - 2 $ to $ x^3 + x + 3 \geq 0 $. Our goal is to find the values of $ x $ where this function is non-negative, i.e., $ f(x) \geq 0 $.

Graph the Function

Using a graphics calculator, graph the function $ f(x) = x^3 + x + 3 $. Make sure the calculator is set to an appropriate window that captures the behavior of the cubic function. Look for changes in sign where the function crosses the x-axis or is tangent to it as these indicate the points where $ f(x) \geq 0 $.

Zoom for Precision

Use the zoom feature on the graphics calculator to focus in on where $ f(x) $ crosses the x-axis. Continue zooming in until the successive x-coordinate values where the crossing or tangency occurs have at least the first three identical digits. This will provide a more precise approximation of the x-coordinates at which $ f(x) = 0 $.

Determine the Solution Set

Once you have derived the approximate x-coordinates where $ f(x) = 0 $, test regions on the interval to determine where $ f(x) \geq 0 $. If necessary, verify by checking additional x-values between known zeros to confirm the sign of $ f(x) $.

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Key Concepts

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

Graphics Calculator

A graphics calculator is an electronic device with the capability to plot graphs, solve equations, and perform sophisticated mathematical calculations. It is particularly helpful when tackling problems involving graphing inequalities. Using a graphics calculator, you can visually interpret where a corresponding function becomes positive or negative by observing the graph. Here's how it helps:

Visualizing complex functions.
Identifying where the function crosses the x-axis.
Offering zoom and trace features for precise calculations.

Leveraging these features to solve inequalities involves plotting the function and examining the areas above (positive) or below (negative) the x-axis.

Cubic Functions

Cubic functions, like the one given by the exercise \[ f(x) = x^3 + x + 3 \]have a degree of three, meaning the graph will have at most three real roots or x-intercepts. Understanding the behavior of cubic functions is crucial. Key characteristics include:

They often exhibit one turning point, creating a series of waves.
They can have up to three real roots.
The graph will tend to infinity in both the positive and negative y-directions.

When tasked with solving cubic inequality problems, break down the function into simpler problems, studying where the function behaves above or below the x-axis.

Zoom Feature

The zoom feature on a graphics calculator is incredibly handy for obtaining precise solutions, especially with inequalities. When working with functions that have gradual shifts or close x-intercepts, if the initial view doesn't provide enough detail, you can use the zoom feature to enhance clarity. Benefits are:

Improves the precision of identifying zeros of functions.
Allows detailed analysis of specific portions of a graph.
Facilitates checking minute changes around critical points.

By continuously zooming into the region of interest, you can refine your understanding of where a cubic function crosses the x-axis with precisions such as matching the first three digits of successive x-values.

Inequality Solving

Inequality solving involves finding where a particular function upholds or violates a given condition, such as being greater than zero. From the exercise, transforming the inequality \[ x^3 + 1 \geq -x - 2 \]to a simpler form like \[ x^3 + x + 3 \geq 0 \]helps pinpoint where the function is non-negative.Steps to solve include:

Graphing the function using a calculator.
Identifying regions above the x-axis.
Testing x-values within intervals between x-intercepts if required.

Applying these methods, supported by calculator technology, enables you to derive a clearer solution set for where the inequality holds true.

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Short Answer

Step by step solution

Define the Function

Graph the Function

Zoom for Precision

Determine the Solution Set

Key Concepts

Graphics Calculator

Cubic Functions

Zoom Feature

Inequality Solving

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