Problem 2 - Elementary Algebra

Question

For the following problems, find the domain of each rational expression. $$ \frac{10 x}{x+6} $$

Step-by-Step Solution

Answer

Answer: The domain of the rational expression is all real numbers except x = -6, which can be written in interval notation as \((-\infty, -6) \cup (-6, \infty)\).

Step 1: Identify the Denominator

In the rational expression, the denominator is the expression below the fraction line, which in this case is \((x + 6)\).

Step 2: Set the Denominator Equal to Zero

To find the values of x for which the expression is undefined, we set the denominator equal to zero and solve for x. $$ x + 6 = 0 $$

Step 3: Solve for x

Solve the equation to find the value of x that makes the denominator equal to zero. $$ x = -6 $$

Step 4: Write the Domain

The domain of the rational expression is all real numbers except the value that makes the denominator zero. In this case, the domain is all real numbers except x = -6. We can write this in interval notation as follows: $$ (-\infty, -6) \cup (-6, \infty) $$

Key Concepts

Denominator

In a rational expression, the denominator is a crucial component. It is essentially the expression that lies beneath the fraction line. For example, in the rational expression \( \frac{10x}{x+6} \), the denominator is \(x + 6\). Identifying the denominator is the first step in examining rational expressions because it affects the domain—the set of permissible input values for the variable. Understanding the denominator's role is critical since any value that makes it zero could result in undefined expressions, and so adjustments are needed. When the denominator equals zero, the fraction does not yield a real number result, which leads us to the next concept of undefined expressions.

Undefined Expressions

An undefined expression in mathematics occurs when an operation produces a result that does not correspond to a specific number. In the context of rational expressions, this primarily happens when the denominator equals zero. Consider the expression \(\frac{10x}{x+6}\). To find when this expression is undefined, we set the denominator equal to zero and solve for \(x\):

• Set \(x + 6 = 0\)
• Solve for \(x\), which gives \(x = -6\).

When \(x = -6\), the expression becomes undefined as it attempts to divide by zero, a mathematical impossibility. Identifying and excluding such values ensures the rational expression uses only valid, defined numbers.

Real Numbers

Real numbers encompass a wide spectrum of numbers that include all possible fractional and decimal values. These are the numbers you encounter every day, like integers, fractions, and decimals. In the context of domain, real numbers refer to all possible values of \(x\) that a rational expression can take, excluding any that make the denominator zero.For the expression \(\frac{10x}{x+6}\), the domain is all real numbers, but we must exclude \(x = -6\) since it makes the expression undefined. Thus, it's crucial to factor out any restrictions from the denominator to identify real numbers for the domain.

Interval Notation

Interval notation is a concise way to express a range of numbers, usually by using parentheses and brackets. It is a useful tool to visually display the domain of functions or expressions. For the rational expression \(\frac{10x}{x+6}\), we determined that it is undefined at \(x = -6\), and thus we exclude this value from the domain.In interval notation, the domain would be expressed as:

• \((-\infty, -6) \, \, (-6, \infty)\)

This notation signifies that any number except \(x = -6\) is permissible. The parenthesis indicates that the endpoint \(-6\) is not included in the set, confirming the exclusion of values that lead to undefined expressions.