Problem 47 - Precalculus

Question

Rewrite each statement using absolute value notation, as in Example 5. The distance between \(x^{2}\) and \(a^{2}\) is less than \(M\).

Step-by-Step Solution

Answer

\(|x^2 - a^2| < M\)

Step 1: Understand the Problem

The problem asks us to express that the distance between two values, \(x^2\) and \(a^2\), is less than \(M\) using absolute value notation. Distance in mathematics between two numbers \(x\) and \(y\) can be written as \(|x - y|\).

Step 2: Apply Absolute Value Notation

To express the distance between \(x^2\) and \(a^2\), we use the absolute value notation to write \(|x^2 - a^2|\). This notation represents the non-negative difference between \(x^2\) and \(a^2\).

Step 3: Set Up the Inequality

Now, we are told that this distance \(|x^2 - a^2|\) is less than \(M\). In mathematical terms, this can be written as the inequality \(|x^2 - a^2| < M\).

Step 4: Final Expression

The final expression using absolute value notation states that the distance between \(x^2\) and \(a^2\) is less than \(M\), which is given by the inequality \(|x^2 - a^2| < M\).

Key Concepts

Distance Between Numbers

In mathematics, distance refers to the absolute difference between two values on a number line. To evaluate the distance between two numbers, we utilize absolute value notation. Absolute value provides a method to account for the magnitude of differences without being impacted by direction.

• Distance is always a non-negative number.
• The distance between numbers ensures you are measuring the gap irrespective of which number is larger.

For instance, the distance between the numbers 5 and 3 can be written as \[|5 - 3| = 2\]Essentially, the expression \(|x - y|\) calculates the distance between numbers \(x\) and \(y\), representing how far apart those values are from each other. This fundamental concept is crucial when establishing relationships and comparisons between values, especially in algebraic expressions.

Inequality Expressions

Inequality expressions are a way to express the relative size or order of two values. Unlike equalities, which state that two values are the same, inequalities show that one is greater than, less than, or not equal to the other.

• Common symbols: \(<\), \(>\), \(\leq\), and \(\geq\).
• Inequality expressions extend beyond simple numbers to include expressions and variables.

In the context of absolute value, an inequality such as \(|x^2 - a^2| < M\) indicates that the distance between \(x^2\) and \(a^2\) is less than some constant \(M\). This represents a boundary or a set limit that defines allowable or expected values. In real-world scenarios, inequalities are used to formulate solutions, constraints, and expectations, proving essential in various fields from science to economics.

Quadratic Expressions

Quadratic expressions are a type of polynomial characterized by an exponent of 2. A standard quadratic expression takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. These expressions are pivotal in many algebraic processes and have distinctive properties:

• They form a parabolic shape when graphed on a coordinate plane.
• The solutions to quadratic equations are found using methods such as factoring, completing the square, or the quadratic formula.
• They can have two, one, or no real solutions based on the discriminant \(b^2 - 4ac\).

In examining the problem, the term \(x^2\) resembles a quadratic form, primarily because it involves a squared term. Recognizing this structure leads to understanding not just simple distances, but how entire polynomial relationships can transform and participate in our analysis of solutions involving quadratic expressions.