Problem 14 - Algebra A Combined Function

Question

Find any numbers for which each rational expression is undefined. $$ \frac{x+1}{5 x-2} $$

Step-by-Step Solution

Answer

The expression is undefined when \( x = \frac{2}{5} \).

Step 1: Understand the Expression

A rational expression is undefined where its denominator is zero. Here, the rational expression is \( \frac{x+1}{5x-2} \). We need to identify the value of \( x \) that makes \( 5x-2 \) equal to zero, which would make the entire expression undefined.

Step 2: Set the Denominator Equal to Zero

Set the denominator of the expression equal to zero and solve for \( x \). The denominator is \( 5x-2 \). Therefore, we need to solve the equation:\[ 5x - 2 = 0 \]

Step 3: Solve the Equation for x

To solve \( 5x - 2 = 0 \), we need to isolate \( x \). Start by adding 2 to both sides of the equation, which gives us: \[ 5x = 2 \]. Then, divide both sides by 5: \[ x = \frac{2}{5} \].

Step 4: Identify the Value that Makes the Expression Undefined

The expression \( \frac{x+1}{5x-2} \) is undefined when \( x = \frac{2}{5} \) because it makes the denominator zero. Therefore, when \( x = \frac{2}{5} \), the expression does not have a valid value.

Key Concepts

Undefined Expressions

When dealing with rational expressions, it's important to know when they become undefined. An expression is undefined in the context of rational expressions if the denominator equals zero. This is because division by zero is mathematically undefined. For instance, if you have the expression \( \frac{x+1}{5x-2} \), it will become undefined at any value of \( x \) that turns the denominator, \( 5x-2 \), into zero.
Imagine you have a fraction and the bottom number suddenly becomes zero - you can't continue the calculation! To find these troublesome values of \( x \), you set the denominator equal to zero and solve.

Denominator

The denominator of a fraction can reveal a lot about the behavior of a rational expression. In the expression \( \frac{x+1}{5x-2} \), '5x-2' is the denominator.
The denominator is key because it determines when the expression is undefined. It's crucial to remember that if the denominator equals zero, the entire fraction becomes meaningless in terms of calculation.
To investigate when this happens, set the denominator equal to zero, which leads us to a solvable equation. In mathematical terms, this is often the first step to understanding when or why a rational expression becomes undefined.

Solving Equations

Solving equations is all about finding the unknown value that makes an equation true. Here, you have the equation \( 5x - 2 = 0 \), which stems from the denominator of our rational expression.
Solving this equation involves reversing operations to isolate \( x \). Each equation-solving step brings you closer to finding the critical point where the denominator fails. Using basic algebra, begin by eliminating any constants added or subtracted from the variable term, and then handle coefficients - numbers multiplying the variable.
Solving equations isn't just about finding any solution but finding the correct one that makes your initial situation mathematically sound.

Isolate Variable

Isolating a variable - like \( x \) in an equation - essentially means getting \( x \) on its own on one side of the equal sign. This is a fundamental skill in algebra that lets you solve for unknowns.
Consider the equation \( 5x - 2 = 0 \). To isolate \( x \), start by eliminating the constant \(-2\) by adding 2 to both sides, resulting in \( 5x = 2 \).
The next step is to deal with the coefficient of 5, which is multiplied by \( x \). To isolate \( x \), divide both sides of the equation by 5, giving: \( x = \frac{2}{5} \).
This value of \( x \) is crucial as it indicates where the expression becomes undefined. The skill of isolating the variable helps identify solutions that clarify the behavior of the expression.