Question

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

Short answer

The domain is \(x eq -4\).

Step-by-step solution

- Understand the Expression

The given expression is \(\frac{x}{x+4}\). We need to find the values of x for which this expression is defined.

- Identify Undefined Points

The expression is a fraction. A fraction is undefined when its denominator is zero. Therefore, we need to find the values of x that make the denominator zero.

- Set Denominator to Zero

Set the denominator of the fraction \(x+4\) to zero and solve for x. So, \(x + 4 = 0\).

- Solve the Equation

Solve the equation \(x + 4 = 0\): \[x = -4\]

- Determine the Domain

Since the expression is undefined when \(x = -4\), we conclude the domain includes all real numbers except \(x = -4\). Hence, the domain is \(x eq -4\).

Key concepts

Fractions

Fractions are an essential concept in mathematics. In a fraction, the number on the top is the numerator, and the number on the bottom is the denominator.
Fractions become more complex when variables are involved, as in the provided exercise.

In general, for a fraction to be defined, the denominator must not be zero. That's why, when dealing with fractions, you should always check for values that make the denominator zero.
This is crucial because dividing by zero is undefined and mathematically invalid.
This brings us to the next core concept—undefined expressions.

Undefined Expressions

An expression becomes undefined when it involves an illegal mathematical operation.
One of the most common cases is dividing by zero as discussed in fractions.

To check for undefined expressions, locate the denominator and set it equal to zero.
Solving this will give you values that make the expression undefined.

In the exercise's example, \( \frac{x}{x+4} \), we identified that setting \( x+4 = 0 \) gives \( x = -4 \).
Therefore, the expression is undefined when \( x = -4 \). This step ensures that the domain excludes these undefined points.

Solving Equations

Solving equations is a fundamental skill in mathematics. It involves finding the value(s) of the variable that make an equation true.

Follow these steps to solve simple linear equations:

• Isolate the variable on one side of the equation.
• Perform arithmetic operations to both sides equally, ensuring the equation remains balanced.
• Check the solution by substituting it back into the original equation.

In the exercise, we solved \( x + 4 = 0 \) by isolating \( x \) and found that \( x = -4 \).
This is a straightforward example, but the same principles apply to more complex equations.