Problem 37 - Elementary and Intermediate Algebra

Question

Find all real numbers for which the rational expression is undefined. See Example 2. $$ \frac{16}{20-x} $$

Step-by-Step Solution

Answer

The expression is undefined when \( x = 20 \).

Step 1: Identify the Rational Expression

The expression given is \( \frac{16}{20-x} \). A rational expression is undefined when its denominator is zero, as division by zero is not allowed.

Step 2: Set the Denominator to Zero

To find when the expression is undefined, set the denominator equal to zero: \[ 20 - x = 0 \].

Step 3: Solve for x

Solve the equation from the previous step to find the values of \( x \) that make the denominator zero:\[ 20 - x = 0 \]\[ -x = -20 \]\[ x = 20 \].

Step 4: Conclusion on Undefined Values

The expression \( \frac{16}{20-x} \) is undefined when \( x = 20 \), because substituting \( 20 \) for \( x \) makes the denominator zero.

Key Concepts

Rational Expressions

A rational expression, such as \( \frac{16}{20-x} \), is essentially a fraction wherein both the numerator and the denominator are polynomials. In simpler terms, it is the division of one polynomial by another. These expressions can take various forms, but the key characteristic is the polynomial structure in the fraction form. A fundamental concept about rational expressions is their domain—the set of all possible values that the variables can take while keeping the expression defined. The most common issue affecting the domain relates closely to division by zero, which is not permissible in mathematics. Therefore, when identifying the domain of any rational expression, it is crucial to determine when the denominator becomes zero and exclude those values.

Division by Zero

Division by zero in any mathematical context leads to an undefined state. This is because dividing any number by zero lacks logical or calculable meaning in standard arithmetic. Consider the rational expression \( \frac{16}{20-x} \); if the denominator \( 20-x \) becomes zero, the whole expression becomes undefined. To avoid any undefined behavior, one must solve for the variable that causes the denominator to reach zero. This is accomplished by setting the entire denominator equal to zero and solving for the variable, as shown:\[ 20 - x = 0 \].

• If \( x = 20 \), the denominator becomes zero, leading to an undefined expression.
• Ensuring that \( x \) does not equal the value that zeros the denominator is crucial for maintaining the rational expression's validity.

Algebraic Equations

Algebraic equations are mathematical statements where two expressions are set equal to each other. They involve variables and constants, and solving them means finding the values of variables that make the equation true. Within the context of our rational expression \( \frac{16}{20-x} \), determining when it's undefined requires solving the equation \( 20 - x = 0 \).
Solving such an equation involves basic algebraic manipulation, where we isolate the variable. For instance, rearranging the terms in \( 20 - x = 0 \) gives us:
\[ -x = -20 \],
which can be simplified further to \( x = 20 \). This result identifies when the original rational expression becomes undefined, showcasing the power of algebra in understanding and solving real-world problems.