Problem 25 - Algebra 1

Question

For what values of the variable is the rational expression undefined? $$\frac{11}{x-8}$$

Step-by-Step Solution

Answer

The rational expression is undefined for \(x = 8\).

Step 1: Identify Denominator

In the given fraction \(\frac{11}{x-8}\), the denominator is \(x - 8\).

Step 2: Set Denominator to Zero

Since a division by zero is undefined in mathematics, set \(x - 8 = 0\).

Step 3: Solve for x

Solving the equation \(x - 8 = 0\) for x, gives us \(x = 8\).

Key Concepts

Denominator in Rational Expressions

When studying rational expressions, it's crucial to understand the role played by the denominator. The denominator of any fraction, which is the bottom part below the division bar, is what we divide the numerator by. In an expression like \(\frac{11}{x-8}\), the denominator is \(x-8\).

It's essential for the denominator to be non-zero because if it equals zero, the entire rational expression becomes undefined. This is why we always look at the denominator first to identify values that could potentially cause a problem. In education, this analysis is fundamental as it lays the groundwork for understanding constraints within functions and solving complex equations.

Solving Equations

Solving equations is a foundational skill in mathematics, essential for success in algebra and beyond. To solve for an unknown variable in an equation, we perform operations that isolate the variable on one side of the equation. This often involves adding, subtracting, dividing, and multiplying both sides of the equation by numbers or expressions until the variable is by itself.

In the context of the exercise given, \(x-8=0\) is a simple linear equation where we solve for \(x\) by adding 8 to both sides, yielding \(x=8\). The understanding that each step taken to solve an equation needs to maintain the balance between both sides is pivotal in achieving accurate results and is applicable in a variety of mathematical disciplines.

Division by Zero

The concept of division by zero is a fundamental 'no-go' in mathematics. In essence, division is breaking a number into a certain amount of equal parts. However, if you attempt to divide by zero, you're essentially trying to break something into zero parts, which does not make sense. This is why any number divided by zero is undefined because there's no way to complete that operation within the realms of real numbers.

Students must be aware that encountering a zero in the denominator is something that must always be avoided. In our exercise, when we set the denominator \(x-8\) equal to zero, this is not because we want to divide by zero, but rather to find the value of \(x\) for which the original expression \(\frac{11}{x-8}\) is undefined. Understanding this concept is crucial because it helps in identifying and preventing nonsensical or impossible solutions in various mathematical contexts.