Problem 44 - Precalculus

Question

Rewrite each statement using absolute value notation, as in Example 5. The distance between \(x^{3}\) and -1 is at most 0.001.

Step-by-Step Solution

Answer

\(|x^3 + 1| \leq 0.001\)

Step 1: Understand the Meanings

First, identify what the statement is asking. The distance between two numbers is typically written using absolute value notation. Here, we want to express that the distance between \(x^3\) and \(-1\) is at most 0.001.

Step 2: Write the General Absolute Value Expression

The absolute value expression for the distance between \(x^3\) and \(-1\) can be expressed as \(|x^3 + 1|\). This represents the magnitude of the difference between \(x^3\) and \(-1\).

Step 3: Set Up the Inequality

Since the distance is at most 0.001, it means that \(|x^3 + 1|\) should be less than or equal to 0.001. Therefore, we write the inequality:\[ |x^3 + 1| \leq 0.001 \]

Step 4: Conclusion

The statement "The distance between \(x^3\) and -1 is at most 0.001" can be rewritten using absolute value notation as:\[ |x^3 + 1| \leq 0.001 \]

Key Concepts

inequalities

Understanding inequalities is crucial when working with absolute value expressions. Inequalities help us define a range of acceptable solutions rather than pinpointing a single answer.
This approach is useful in numerous contexts across mathematics. For absolute values, inequalities often determine how close or far a number can be from a specific point.

• When we have \(|a| \leq b\), it means the value of \(|a|\) is not larger than \(b\).
• The expression can be split into two parts: \(a \leq b\) and \(-a \leq b\).

In the original problem, the inequality \[ |x^3 + 1| \leq 0.001 \] indicates how much the expression \(x^3\) deviates from \(-1\).
By understanding this inequality, we can determine that \(x^3\) must remain within a tight boundary, ensuring the deviation does not exceed 0.001.
This usage of inequalities in conjunction with absolute values is a powerful tool for capturing precise relationships between numbers.

distance between numbers

The concept of distance between numbers is akin to measuring the gap between two points on a numerical line. Absolute value helps simplify this idea by focusing only on the size of that gap, without concern for direction.
When we talk about the distance between \(x^3\) and \(-1\), we want to know how far these two values are from each other.

• Absolute value shows the non-negative distance, represented as \(|x^3 + 1|\) here.
• It doesn't matter whether \(x^3\) is greater or smaller than \(-1\). The absolute value removes the sign and gives us a straightforward measure.

In mathematics, representing the distance using absolute value is common practice.
This ensures a consistent method to address real-life scenarios where only the magnitude matters, such as measuring physical distances or calculating error tolerances.

notation

Notation in mathematics serves as a compact language that communicates complex ideas clearly and effectively. Absolute value notation is a brilliant example of this, providing a visual and mathematical way to depict the size of a number regardless of its sign.
In the problem, \[ |x^3 + 1| \leq 0.001 \] is not just a collection of symbols but a language tool that conveys crucial information in a concise manner.

• \(|x^3 + 1|\) means "the absolute value of \(x^3 + 1\)," indicating the distance from zero on a number line.
• The inequality sign \(\leq\) shows a boundary: the distance should not exceed 0.001.

This concise form of notation allows mathematicians and students to quickly understand and work with complex concepts without lengthy descriptions.
Understanding how to read, write, and apply mathematical notation is essential to tackling more advanced mathematical concepts effectively.