Problem 15 - Elementary and Intermediate Algebra

Question

Evaluate each expression for \(x=6 .\) See Example 1. $$ \frac{-x+1}{x^{2}-5 x-6} $$

Step-by-Step Solution

Answer

The expression is undefined for x = 6.

Step 1: Substitute x=6 into the numerator

The given expression is \( \frac{-x+1}{x^{2}-5x-6} \). First, we substitute \( x = 6 \) into the numerator which is \( -x + 1 \). Therefore, it becomes \( -(6) + 1 = -6 + 1 = -5 \).

Step 2: Substitute x=6 into the denominator

Now substitute \( x = 6 \) into the denominator which is \( x^{2} - 5x - 6 \). So it becomes \( (6)^2 - 5(6) - 6 \). Calculate each term: \( 6^2 = 36 \), \( 5 \times 6 = 30 \). Therefore the expression becomes: \( 36 - 30 - 6 \).

Step 3: Simplify the denominator

Simplify \( 36 - 30 - 6 \) to find the denominator value: First \( 36 - 30 = 6 \), then \( 6 - 6 = 0 \). So the denominator equals zero.

Step 4: Evaluate the expression

The expression now is \( \frac{-5}{0} \). Division by zero is undefined in mathematics. Therefore, the given expression is undefined for \( x = 6 \).

Key Concepts

Substitution Method

The substitution method is a key technique in algebra used to evaluate expressions by replacing variables with their given numerical values. In this exercise, we replaced the variable \( x \) with the value 6 in both the numerator and the denominator of the expression. The process involves:

• Identifying the variable in the expression.
• Replacing each instance of the variable with the specified number.
• Simplifying where necessary to find the resultant value.

For example, the numerator part \( -x + 1 \) becomes \( -6 + 1 \), simplifying to \(-5\). The denominator \( x^2 - 5x - 6 \) is replaced with \( 6^2 - 5(6) - 6 \). This eventually simplifies to \( 36 - 30 - 6 \). Substituting helps in transforming algebraic expressions into simpler arithmetic operations. It's a straightforward method favored for its simplicity and effectiveness, providing a more tangible way to handle abstract algebraic concepts.

Division by Zero

Dividing any number by zero is a fundamental concept that often puzzles learners. In mathematics, division by zero is undefined because it leads to unpredictable outcomes. To understand why let's consider the expression \( \frac{-5}{0} \). Attempting to divide by zero can break basic arithmetic rules. Mathematically, division is essentially determining how many times a divisor fits into a dividend. But with zero, this leads to logical inconsistencies:

• No sensible number of divisions fit zero into a non-zero number.
• It's impossible to multiply zero by any finite number to get back the original dividend, since zero times anything is zero.
• This is why computing \( \frac{-5}{0} \) doesn't result in a meaningful number.

Avoid dividing by zero in calculations and be aware that expressions requiring this operation are termed undefined.

Undefined Expressions

An undefined expression occurs when an operation in algebra cannot be reasonably or logically computed, as seen when division by zero takes place. In our scenario, substituting \( x = 6 \) turned the denominator into zero, making the whole expression \( \frac{-5}{0} \) undefined. Here's why understanding undefined expressions matter:

• They often indicate limits where a function isn't valid.
• It's essential for problem solving to recognize situations leading to undefined outcomes to reinterpret or reconsider methods.
• The occurrence of undefined expressions teaches about the domain and limitations of functions and mathematical rules.

Undefined expressions prompt the need for checking domains and ensure methods like substitution can be applied properly. Always scrutinize algebraic results to confirm each part of a problem is mathematically sound.