Question

Determine the domain of the variable \(x\) in each expression. \(\frac{-6}{x+4}\)

Short answer

The domain of \( \frac{-6}{x+4} \) is \ x eq -4 \.

Step-by-step solution

Identify the Expression

Examine the given mathematical expression: \ \( \frac{-6}{x+4} \ \).

Determine Restrictions

To find the domain, identify any values of \(x\) that would make the denominator zero. Set the denominator equal to zero: \( x + 4 = 0 \).

Solve the Equation

Solve the equation from Step 2 to find the restricted value of \(x\): \ \( x + 4 = 0 \ \Rightarrow x = -4 \ \).

State the Domain

The domain includes all real numbers except the restricted value found in Step 3. Therefore, the domain of \(x\) is all real numbers except \(x = -4\).

Key concepts

Understanding Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. In simpler terms, it's like having a division problem where the top and the bottom parts are algebraic expressions involving variables.
Diving into the given exercise, we have the rational expression \( \frac{-6}{x+4} \). Here, \( -6 \) is the numerator, and \( x + 4 \) is the denominator.
Whenever dealing with rational expressions, it's crucial to be mindful of the denominator. This is because the denominator must never be zero. A zero denominator makes the expression undefined, and thus, we need to find values of the variable that would lead to this issue.

Domain Restrictions in Rational Expressions

The domain of an expression is the set of all possible values that the variable (in this case, \( x \)) can take. While rational expressions can often take a broad range of values, we need to be extra cautious about the denominator.
In our example, the expression is \( \frac{-6}{x+4} \). To ensure that our expression is always defined, we must find and exclude values of \( x \) that turn the denominator into zero.
For the denominator \( x + 4 \), setting it equal to zero helps identify these restricted values:
\[ x + 4 = 0 \]
Solve this equation, and you'll find:
\[ x = -4 \]
This tells us that \( x = -4 \) is not allowed. Hence, the domain is all real numbers except \( x = -4 \).

Why a Zero Denominator is a Problem

When the denominator of a rational expression equals zero, the fraction becomes undefined. This is because division by zero is not possible in mathematics.
To illustrate, let's refer back to our expression \( \frac{-6}{x+4} \). If we substitute \( x = -4 \) in the denominator, we get:
\[ \frac{-6}{-4 + 4} = \frac{-6}{0} \]
Since division by zero is undefined, \( x = -4 \) can't be part of the domain.
Always ensure that the denominator is never zero, which involves careful inspection and exclusion of specific values of the variable. This step is essential in determining the valid domain of the expression.
In conclusion, the domain of the variable \( x \) in the given rational expression includes all real numbers except the value that makes the denominator zero.