Identify the property illustrated in each statement. Assume all variables represent real numbers. (t - 6) • ( 1/(t-6)) = 1, if t - 6 ≠ 0

Verified step by step guidance

1. Observe the expression given: \((t - 6) \cdot \left( \frac{1}{t - 6} \right) = 1\), with the condition that \(t - 6 \neq 0\).
2. Recall the property of multiplication involving a number and its reciprocal: for any nonzero number \(a\), \(a \cdot \frac{1}{a} = 1\).
3. Identify that in this case, \(a\) corresponds to the expression \((t - 6)\), and its reciprocal is \(\frac{1}{t - 6}\).
4. Since multiplying a number by its reciprocal results in 1, this equation illustrates the Multiplicative Inverse Property.
5. Therefore, the property shown here is the Multiplicative Inverse Property, which states that any nonzero number multiplied by its reciprocal equals 1.

Key Concepts

Multiplicative Inverse Property

This property states that for any nonzero real number a, multiplying a by its reciprocal (1/a) results in 1. It is fundamental in solving equations and simplifying expressions involving division.

Nonzero Condition in Division

Division by zero is undefined, so the denominator in a fraction must be nonzero. This condition ensures the expression is valid and the multiplicative inverse property can be applied.

Properties of Real Numbers

Real numbers follow specific algebraic rules, including the existence of multiplicative inverses for all nonzero elements. Understanding these properties helps in manipulating and simplifying expressions correctly.