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Chapter 0: Problem 90

In Exercises 85-96, identify the rule(s) of algebra illustrated by the statement. $$ (z-2)+0=z-2 $$

Short Answer

Expert verified
The rule of algebra illustrated by the statement is the Additive Identity Property.

Step by step solution

01

Understand the given equation

Compare the left hand side (LHS) of the equation with the right hand side (RHS). The LHS is \( (z-2)+0 \) while the RHS is\( z-2 \).
02

Analyze the operation

Observe that the LHS includes addition of zero to the term \( z-2 \).
03

Identify the algebraic property

Identify the operation as an application of the Additive Identity Property which states that any number plus zero equals the original number. The statement shows that when zero is added to \( z-2 \), the result is still \( z-2 \). So the algebraic property illustrated by the given mathematical statement is the Additive Identity Property.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Properties
Algebraic properties are fundamental rules that govern how arithmetic operations can be performed on numbers. These rules are incredibly helpful in simplifying and solving equations.

Some commonly known algebraic properties include the commutative, associative, distributive, and identity properties. Each of these plays a specific role in mathematical problem-solving. They help us understand how numbers interact with each other under different operations.

The additive identity property, as we see in the solution from the exercise, is an example of an algebraic property. This specific property shows how adding zero affects numbers in algebra.
Addition in Algebra
Addition in algebra is similar to simple arithmetic addition, but it often involves variables and constants. The primary goal is to manage and combine terms.

When performing addition in algebra, it is crucial to follow established properties, such as the commutative property, which allows us to switch the order of terms鈥攖he sum remains unchanged. Also, terms that have the same variables can be combined to simplify expressions further.

In algebra, understanding concepts like terms and coefficients is essential. Only like terms, which are terms with the same variable raised to the same power, can be added together. Thus, efficiently combining like terms and utilizing the additive identity property aids in solving algebraic expressions easily.
Zero in Mathematics
Zero is a unique number in mathematics with significant importance. It acts as a placeholder in our number system and is crucial in arithmetic operations.

One of the most interesting roles of zero is its behavior in addition. According to the additive identity property, any number added to zero remains unchanged. Therefore, zero is referred to as the additive identity.

Zero's significance extends beyond simple arithmetic. In algebra, it is used to balance equations and is frequently a solution when solving linear equations. Additionally, zero serves as the reference point on the number line, which helps in dividing numbers into positive and negative categories. Understanding how zero interacts with numbers enriches overall mathematical comprehension.

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Most popular questions from this chapter

Maximum Profit The cost per unit in the production of a portable CD player is \(\$ 60\). The manufacturer charges \(\$ 90\) per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by \(\$ 0.15\) per CD player for each unit ordered in excess of 100 (for example, there would be a charge of \(\$ 87\) per CD player for an order size of 120 ). (a) The table shows the profit \(P\) (in dollars) for various numbers of units ordered, \(x\). Use the table to estimate the maximum profit. \begin{tabular}{|l|c|c|c|c|} \hline Units, \(x\) & 110 & 120 & 130 & 140 \\ \hline Profit, \(P\) & 3135 & 3240 & 3315 & 3360 \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|} \hline Units, \(x\) & 150 & 160 & 170 \\ \hline Profit, \(P\) & 3375 & 3360 & 3315 \\ \hline \end{tabular} (b) Plot the points \((x, P)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(P\) as a function of \(x\) ? (c) If \(P\) is a function of \(x\), write the function and determine its domain.

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{x-3}{x+2} $$

Each function models the specified data for the years 1995 through 2005 , with \(t=5\) corresponding to 1995 . Estimate a reasonable scale for the vertical axis (e.g., hundreds, thousands, millions, etc.) of the graph and justify your answer. (There are many correct answers.) (a) \(f(t)\) represents the average salary of college professors. (b) \(f(t)\) represents the U.S. population. (c) \(f(t)\) represents the percent of the civilian work force that is unemployed.

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{5}-2 $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{4}{x} $$
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