Problem 34 - Intermediate Algebra

Question

Complete each statement so that the indicated property is illustrated. See Example 1. \(h+(-h)=______ \) Additive inverse property

Step-by-Step Solution

Answer

The answer is 0.

Step 1: Identify the Property

The exercise refers to the 'Additive Inverse Property'. This property states that the sum of a number and its additive inverse equals zero. In mathematical terms, for any real number \(a\), \(a + (-a) = 0\).

Step 2: Apply the Additive Inverse Property

We are given the expression \(h + (-h)\). According to the additive inverse property, adding a number and its additive inverse results in zero. Therefore, \(h + (-h) = 0\).

Step 3: Complete the Statement

To illustrate the property in the original mathematical expression, fill in the blank with the correct value: \(h + (-h) = 0\).

Key Concepts

Real Numbers

Real numbers encompass a vast set of numbers that include various different types encountered in basic mathematics. Real numbers can be divided into rational and irrational numbers.

• Rational numbers: These are numbers that can be expressed as the quotient of two integers. For example, \( \frac{3}{4} \), 2, and -5 are all rational numbers.
• Irrational numbers: These cannot be expressed as simple fractions. Examples include \( \pi \) (Pi) and \( \sqrt{2} \) (the square root of 2).

The concept includes whole numbers, natural numbers, and integers as well. Each type falls under the umbrella of real numbers. This concept is fundamental as it’s often used in solving algebraic equations and understanding mathematical functions. By recognizing different types of real numbers, students can better grasp operations like addition, which involves their additive inverses.

Additive Inverse

The additive inverse is a fundamental concept in basic algebra. For each real number, there is another number called its additive inverse. The key idea here is that a number and its additive inverse sum up to zero.For any real number \(a\):

• The additive inverse is \(-a\).
• When you add \(a\) and \(-a\), the result is always 0, as shown by the equation \(a + (-a) = 0\).
• This property can help solve equations and simplify expressions by eliminating terms.

In practical situations, the additive inverse can be understood through simple day-to-day examples. Consider if you have \\(5 (represented as \(5\)) and owe a debt of \\)5 (represented as \(-5\)). Adding these together (\(5 + (-5)\)) results in 0, reinforcing that the addends completely cancel each other out.

Basic Algebra

Basic algebra is an introduction to the world of mathematics where numbers and operations meet letters and symbols. It involves foundational concepts such as variables, constants, and operations.

• Variables: These are represented by letters (e.g., \( h \), \( x \), \( y \)) and can take various numerical values. Variables allow us to create general formulas and expressions applicable in numerous situations.
• Constants: These are fixed numbers (e.g., 2, -3) that do not change.
• Basic operations: This includes addition, subtraction, multiplication, and division. Each operation functions with both variables and constants in various expressions and equations.

In the context of additive inverses and expressions such as \( h + (-h) = 0 \), basic algebra provides a framework for understanding how operations can simplify or solve equations. Recognizing how variables work within an expression helps students effectively engage with more complex problems in mathematics, setting the stage for higher-level math courses.