Question

Which of the following is undefined? A. \(\frac{-2+2}{8+(-4)}\) B. \(\frac{5+(-2)}{-7+7}\) C. \(\frac{4-(-1)}{6-3}\) D. \(\frac{-3+(-1)}{4-(-6)}\)

Short answer

Option B is undefined.

Step-by-step solution

Evaluate the numerator and denominator for Option A

For Option A, the expression is \(\frac{-2+2}{8+(-4)}\). Calculate the numerator: \(-2 + 2 = 0\). Now, calculate the denominator: \8 + (-4) = 4\. The fraction becomes \(\frac{0}{4}\), which is defined as 0.

Evaluate the numerator and denominator for Option B

For Option B, the expression is \(\frac{5+(-2)}{-7+7}\). Calculate the numerator: \5 + (-2) = 3\. Now, calculate the denominator: \-7 + 7 = 0\. The fraction becomes \(\frac{3}{0}\), which is undefined because division by zero is not allowed.

Evaluate the numerator and denominator for Option C

For Option C, the expression is \(\frac{4-(-1)}{6-3}\). Calculate the numerator: \4 - (-1) = 5\. Now, calculate the denominator: \6 - 3 = 3\. The fraction becomes \(\frac{5}{3}\), which is defined.

Evaluate the numerator and denominator for Option D

For Option D, the expression is \(\frac{-3+(-1)}{4-(-6)}\). Calculate the numerator: \-3 + (-1) = -4\. Now, calculate the denominator: \4 - (-6) = 10\. The fraction becomes \(\frac{-4}{10}\), which is defined.

Conclusion

After evaluating all options, the only expression that is undefined is Option B because it involves division by zero.

Key concepts

Division by Zero

Division by zero is a critical concept in mathematics. When we talk about dividing by zero, we mean any fraction where the denominator is zero, like \(\frac{1}{0}\). This is undefined because there is no number that you can multiply by 0 to get 1. Trying to divide by zero leads to an undefined result, as it's impossible to partition something into zero parts.
Dividing any number by zero does not have a meaningful outcome in the realm of real numbers. It is an invalid operation, and it’s fundamental to recognize this in algebra.

Numerators and Denominators

In a fraction, the numerator and denominator are crucial components. The numerator is the top part of the fraction and represents how many parts we have. The denominator is the bottom part and shows into how many parts the whole is divided.
For example, in \(\frac{3}{4}\), 3 is the numerator indicating three parts, and 4 is the denominator indicating the entire amount is divided into four parts.
Understanding these terms helps when evaluating mathematical expressions involving fractions.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
For instance, for the fraction \(\frac{10}{20}\), the GCD of 10 and 20 is 10. So, you divide both the numerator and the denominator by 10: \(\frac{10 \div 10}{20 \div 10} = \frac{1}{2}\).
Simplifying helps in evaluating and comparing fractions more easily.

Evaluating Expressions

When evaluating expressions, it's important to follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that you get the correct result.
For instance, in the expression \(\frac{4-(-1)}{6-3}\), first handle the operations inside the parentheses: \4-(-1) = 5\ and \6-3=3\. So, the expression simplifies to \(\frac{5}{3}\).
Evaluating expressions accurately is essential in solving algebraic problems.