Problem 24 - Algebra 1

Question

For what values of the variable is the rational expression undefined? $$\frac{7}{x-3}$$

Step-by-Step Solution

Answer

The rational expression is undefined when x equals 3.

Step 1: Identify the Denominator

First, identify the denominator of the rational expression. In this case the denominator is \(x - 3\).

Step 2: Set the Denominator Equal to Zero

Next, set the denominator equal to zero and solve for the variable x. The expression is: \(x - 3 = 0\).

Step 3: Solve for Variable x

Now solve the equation for x: If \(x - 3 = 0\), then \(x = 3\) after adding 3 to both sides of the equation.

Key Concepts

Undefined Expressions

Rational expressions can sometimes be undefined. This happens when the denominator equals zero, since division by zero is not allowed in mathematics. Whenever you encounter a rational expression, such as \( \frac{7}{x-3} \), it is important to determine the values of the variable that make it undefined.

To find these values, follow this simple process:

• Identify the denominator of the expression.
• Set the denominator equal to zero.
• Solve the resulting equation for the variable.

In this case, solving \( x - 3 = 0 \) helps us find that the rational expression is undefined when \( x = 3 \). This means for any input of \( x = 3 \), the expression does not exist.

Denominator

The denominator is the bottom part of a fraction or rational expression in the form \( \frac{a}{b} \). It is crucial because it indicates how many equal parts the whole is divided into. If the denominator equals zero, the expression becomes undefined, as division by zero is impossible.Finding the denominator in a rational expression:

• Look directly below the division line in the expression to identify it.
• In \( \frac{7}{x-3} \), the denominator is \( x - 3 \).

By understanding the denominator, we can prevent undefined expressions and make sense of when rational expressions are meaningful.

Solving Equations

Solving equations is a fundamental skill in mathematics. It's used to find the values of variables that make an equation true. In the context of rational expressions, solving equations helps determine when these expressions become undefined. Here's a step-by-step guide on solving the equation to find these values:

• Take the denominator of the expression and set it equal to zero.
  For \( x - 3 = 0 \).
• Use basic algebraic operations to solve the equation.
  In this example, add 3 to both sides: \( x = 3 \).

By correctly solving the equation, you identify that the expression \( \frac{7}{x-3} \) is undefined at \( x = 3 \). This approach helps avoid errors in calculations by highlighting potential issues with the expression.