Problem 10 - Algebra A Combined Function

Question

Find any numbers for which each rational expression is undefined. $$ \frac{3}{5 x} $$

Step-by-Step Solution

Answer

The expression is undefined for \( x = 0 \).

Step 1: Understand when a rational expression is undefined

A rational expression is undefined when its denominator is equal to zero. Therefore, to find when the expression \( \frac{3}{5x} \) is undefined, we need to determine when the denominator, \( 5x \), is zero.

Step 2: Set the denominator equal to zero

To find the values of \( x \) that make the denominator zero, set the expression \( 5x = 0 \). This equation helps us find the value of \( x \) for which the expression becomes undefined.

Step 3: Solve for x

Solve the equation \( 5x = 0 \) by dividing both sides by 5 to isolate the variable \( x \). This gives us \( x = \frac{0}{5} = 0 \).

Step 4: Conclusion

The rational expression is undefined when \( x = 0 \). Therefore, substituting \( x = 0 \) into the denominator indeed confirms that the denominator becomes zero, thus making the rational expression undefined.

Key Concepts

Undefined Expressions

In the world of mathematics, especially when dealing with rational expressions, the concept of undefined expressions is crucial. A rational expression is similar to a fraction in that it has a numerator and a denominator. However, it is undefined whenever the denominator equals zero. This is because dividing by zero is not possible in standard arithmetic.

• A rational expression such as \( \frac{3}{5x} \) becomes undefined if its denominator, \( 5x \), equals zero.
• Generally, the undefined state arises when there is any value that, once substituted into the denominator, results in zero.

Understanding when expressions become undefined helps us identify potential problem areas in equations involving variables. By realizing this, we can focus on dealing with such issues in other calculations.
Remember, identifying undefined expressions is not only about finding values that don’t work but knowing how these values impact the solution and structure of mathematical problems.

Denominator Zero

The concept of the denominator equaling zero is central to finding undefined expressions in rational equations. Imagine driving towards a cliff edge — reaching a zero in the denominator is like reaching that edge. You need to know when it will happen to avoid a mathematical pitfall.

• In the case of \( \frac{3}{5x} \), setting the denominator \( 5x \) equal to zero highlights where the expression becomes undefined.
• Mathematically, \( 5x = 0 \) is our starting equation to figure out these critical values.

To tackle this, we solve \( 5x = 0 \), which tells us that \( x \) must be zero for the denominator to "disappear" into an undefined state.
Ensuring the denominator is never zero allows rational expressions to remain valid, ensuring smooth algebraic operations. Thus, identifying these points can prevent errors in larger computations.

Solving Equations

Solving equations where the goal is to find values that make a rational expression undefined involves isolating the variable in the denominator. Let's look into the equation solving process:

• We begin with identifying the expression's denominator — for \( \frac{3}{5x} \), this is \( 5x \).
• Set \( 5x = 0 \) because we want to know when the denominator becomes zero.

The process involves simple algebraic steps:

• Divide both sides of the equation \( 5x = 0 \) by 5, leading to \( x = \frac{0}{5} \).
• Simplifying gives \( x = 0 \).

This single value of \( x \) makes the entire expression undefined. The equation solving method—setting denominators to zero—alerts us to potential undefined values. It's a basic yet vital skill in algebra, allowing safe navigation through complex expressions and maintaining correctness in solutions.