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

Verified step by step guidance

1. Recognize that the expression \((t - 6) \cdot \left(\frac{1}{t - 6}\right) = 1\) is an example of a property of multiplication.
2. Identify that the property being used here is the 'Multiplicative Inverse Property'.
3. Understand that the Multiplicative Inverse Property states that for any non-zero number \(a\), \(a \cdot \frac{1}{a} = 1\).
4. Note that in this problem, \(a\) is represented by \(t - 6\), and \(\frac{1}{a}\) is \(\frac{1}{t - 6}\).
5. Conclude that the statement illustrates the Multiplicative Inverse Property, which holds true as long as \(t - 6 \neq 0\) to avoid division by zero.

Key Concepts

Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, at least one of the factors must be zero. In the context of the given statement, it implies that if the expression (t - 6) multiplied by another expression equals 1, then (t - 6) cannot be zero, ensuring that the multiplication is valid and the equation holds true.

Identity Property of Multiplication

The Identity Property of Multiplication states that any number multiplied by one remains unchanged. In the statement, the expression (t - 6) • (1/(t - 6)) = 1 illustrates this property, as multiplying (t - 6) by its reciprocal yields the identity element, which is 1, provided that (t - 6) is not zero.

Reciprocal

A reciprocal of a number is defined as 1 divided by that number. In the context of the question, the expression (1/(t - 6)) serves as the reciprocal of (t - 6). This concept is crucial for understanding how the multiplication of a number by its reciprocal results in the identity element, reinforcing the relationship between the two expressions in the equation.