Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

0. Review of College Algebra

Solving Linear Equations

Trigonometry

0. Review of College Algebra

Solving Linear Equations

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1:31 minutes

Problem 107

Textbook Question

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

Recognize that the expression \((t - 6) \cdot \left(\frac{1}{t - 6}\right) = 1\) is an example of a property of multiplication.

Identify that the property being used here is the 'Multiplicative Inverse Property'.

Understand that the Multiplicative Inverse Property states that for any non-zero number \(a\), \(a \cdot \frac{1}{a} = 1\).

Note that in this problem, \(a\) is represented by \(t - 6\), and \(\frac{1}{a}\) is \(\frac{1}{t - 6}\).

Conclude that the statement illustrates the Multiplicative Inverse Property, which holds true as long as \(t - 6 \neq 0\) to avoid division by zero.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.