Question

Determine the domain of the variable \(x\) in each expression. \(\frac{x-2}{x-6}\)

Short answer

The domain is \( (-\infty, 6) \cup (6, \infty) \).\

Step-by-step solution

- Identify the restriction

The given expression is \(\frac{x-2}{x-6}\). To determine the domain, the denominator must not be zero, as division by zero is undefined.

- Set the denominator to zero

Set the denominator of the fraction equal to zero and solve for \(x\): \ x - 6 = 0 \.

- Solve the equation

Solve the equation for \(x\): \ x = 6 \. This is the value that will make the denominator zero and is therefore excluded from the domain.

- Write the domain

Since \(x = 6\) is excluded, the domain includes all real numbers except \(x = 6\). In interval notation, the domain is written as \((-\infty, 6) \cup (6, \infty)\).\

Key concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (addition, subtraction, multiplication, and division). For example, in the expression \(\frac{x-2}{x-6}\), \(x \) is the variable. Algebraic expressions are used frequently in algebra to represent real-world problems and abstract mathematical concepts. Understanding and manipulating these expressions is crucial for solving equations and inequalities.

Denominator Exclusion

When working with fractions, it is critical to ensure the denominator is not zero, since division by zero is undefined. To determine the values that must be excluded from the domain, we separate the denominator and set it equal to zero.
In the expression \(\frac{x-2}{x-6}\), we set the denominator \(x-6\) equal to zero:
\[ x - 6 = 0 \]
Solving for \(x\), we get \[ x = 6 \]
Therefore, \(x = 6\) must be excluded from the domain to avoid division by zero.

Interval Notation

Interval notation is a way to represent a set of numbers between two endpoints, and it is often used to describe the domain and range of functions. The most common interval notations include:

• Open intervals: \( (a, b) \) indicates all numbers between \(a\) and \(b\), but not including \(a\) and \(b\).
• Closed intervals: \( [a, b] \) includes all numbers between, and including, \(a\) and \(b\).
• Combinations: \( (a, b] \) and \( [a, b) \) include some bounds but not others.

For our problem, we exclude \(x = 6\) from the domain. So, we use a combination of open intervals to express that all real numbers, except \(6\), are included. The domain is written as \((-\ufffd, 6) \cup (6, \infty)\).