Ch. P - Fundamental Concepts of Algebra

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 83

Chapter 1, Problem 83

State the name of the property illustrated. 1/(x+3) (x+3)=1, x≠−3

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Identify the expression given: $\frac{1}{x+3} \cdot (x+3) = 1$, with the restriction $x \neq -3$ to avoid division by zero.

Recognize that multiplying a number by its reciprocal results in 1. Here, $\frac{1}{x+3}$ is the reciprocal of $(x+3)$.

This illustrates the property that any nonzero number multiplied by its multiplicative inverse (reciprocal) equals 1.

State the property name: This is the Multiplicative Inverse Property.

Note the importance of the restriction $x \neq -3$ to ensure the denominator is not zero, which is necessary for the property to hold.

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Key Concepts

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

Multiplicative Inverse Property

This property states that for any nonzero number a, multiplying it by its reciprocal (1/a) results in 1. In the given expression, (1/(x+3)) multiplied by (x+3) equals 1, illustrating that (x+3) and its reciprocal are multiplicative inverses, provided x ≠ -3 to avoid division by zero.

Domain Restrictions

Domain restrictions specify values that variables cannot take to avoid undefined expressions. Here, x ≠ -3 because substituting -3 would make the denominator zero, which is undefined in algebra. Recognizing these restrictions is essential to correctly apply properties and solve equations.

Properties of Rational Expressions

Rational expressions are fractions involving polynomials. Understanding how to simplify, multiply, and identify restrictions in rational expressions is key. The problem demonstrates multiplying a rational expression by its denominator, simplifying to 1, which relies on these properties.

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