Problem 2 - Precalculus : Building Concepts and Connections

Question

For which values of \(x\) is the following rational expression defined? $$ \frac{x+2}{(x-1)(x+5)} $$

Step-by-Step Solution

Answer

The given rational expression is undefined when \(x = 1\) and \(x = -5\).

Step 1: Set the denominator equal to zero

To figure out which values of \(x\) make the expression undefined, we need to set the denominator equal to zero, since division by zero is undefined in mathematics. Thus, set \((x-1)(x+5) = 0\).

Step 2: Solve for x

This equation can be factored as two separate equations because of the multiplication. The equations are: \(x - 1 = 0\) and \(x + 5 = 0\). Solving these gives \(x = 1\) and \(x = -5\).

Step 3: Conclusion

Therefore, the given rational expression will be undefined when \(x = 1\) and \(x = -5\). This is because these values of \(x\) make the denominator of the fraction equal to zero. In all other cases, the expression is well-defined.

Key Concepts

Division by Zero

In mathematics, division by zero is an operation that is not allowed. This is because division typically involves splitting something into smaller parts.
But dividing by zero doesn't relate to any meaningful action.
For instance, consider dividing 10 by 2: you get two groups of 5. However, what happens when you divide by zero? There are no parts to split into.
Imagine trying to split a cake among zero people—it doesn't make sense.
In rational expressions, such as \( \frac{x+2}{(x-1)(x+5)} \), the rule is the same. If the denominator equals zero, the expression becomes undefined. Therefore, we first need to determine when \( (x-1)(x+5) = 0 \).
The values that make this true result in division by zero, rendering the expression undefined.

Undefined Expressions

Undefined expressions occur when we fall into a situation that defies the rules of mathematics, such as dividing by zero.
When working with rational expressions like \( \frac{x+2}{(x-1)(x+5)} \), you must check the denominator.
This ensures you're not accidentally creating an undefined expression.
The critical step is finding out when the denominator equals zero.
If \((x-1)(x+5) = 0\), solving \(x-1=0\) and \(x+5=0\) gives us \(x=1\) and \(x=-5\).
These are the values where the expression becomes undefined because they make the denominator zero.
In contrast, for all other values of \(x\), the expression will remain valid.

Factoring Equations

Factoring is a mathematical method used to simplify expressions and solve equations.
In rational expressions, it's especially useful for finding when possible undefined issues occur.
Let's look at \((x-1)(x+5) = 0\), the denominator of our expression.
Using factoring, we can understand when each part of the equation equals zero.

• First, consider \(x - 1 = 0\). Solving for \(x\) gives \(x = 1\).
• Next, consider \(x + 5 = 0\). Solving for \(x\) gives \(x = -5\).

These solutions tell us exactly when the denominator will be zero, leading to an undefined expression.
Factoring makes it easy to identify these critical points, helping to avoid mistakes in calculations and understand when expressions might be misleading.