Problem 28 - Algebra 2

Question

Identify the additive inverse and multiplicative inverse for each number. $$ -10 $$

Step-by-Step Solution

Answer

Additive inverse: 10; Multiplicative inverse: \(-\frac{1}{10}\).

Step 1: Understand the Concept of Additive Inverse

The additive inverse of a number is what you add to that number to get zero. In other words, for a number \( x \), its additive inverse is \( -x \).

Step 2: Identify Additive Inverse of -10

To find the additive inverse of \(-10\), we need to determine a number that adds to \(-10\) to result in zero. Mathematically, this is: \(-10 + ? = 0\). The solution is \( 10 \), meaning the additive inverse of \(-10\) is \( 10 \).

Step 3: Understand the Concept of Multiplicative Inverse

The multiplicative inverse (or reciprocal) of a number is what you multiply by that number to get one. For a non-zero number \( x \), its multiplicative inverse is \( \frac{1}{x} \).

Step 4: Identify Multiplicative Inverse of -10

To find the multiplicative inverse of \(-10\), we need to find a number that when multiplied by \(-10\) results in one. Mathematically, this is: \(-10 \cdot ? = 1\). The solution is \( -\frac{1}{10} \), meaning the multiplicative inverse of \(-10\) is \( -\frac{1}{10} \).

Key Concepts

Additive Inverse

The additive inverse is one of the foundational elements of algebra.An additive inverse of a number is what you would add to that number to make the sum zero.For any given number \( x \), its additive inverse is \(-x\).This concept is crucial because it helps in solving equations by allowing terms to cancel out.When you add a number to its additive inverse, the result is always zero.For example:

• Additive inverse of \(5\) is \(-5\), and \(5 + (-5) = 0\).
• For \(-10\), its additive inverse is \(10\), because \(-10 + 10 = 0\).

Understanding additive inverse is essential when you want to isolate variables or simplify algebraic expressions.

Multiplicative Inverse

The multiplicative inverse, often called the reciprocal, is another key concept in algebra.It is what you multiply by a number to achieve the product of one.For a non-zero number \( x \), the multiplicative inverse is \( \frac{1}{x} \).This is because multiplying \( x \) and \( \frac{1}{x} \) gives \( 1 \).The concept is incredibly useful in division and equation solving:

• For example, the multiplicative inverse of \(4\) is \( \frac{1}{4} \), as \(4 \times \frac{1}{4} = 1\).
• In the case of \(-10\), its multiplicative inverse is \(-\frac{1}{10}\), because \(-10 \times -\frac{1}{10} = 1\).

This helps in solving equations where you need to isolate a variable by dividing both sides by the same number.

Reciprocal

The reciprocal of a number is simply another name for its multiplicative inverse.Every non-zero number has a reciprocal, which is obtained by flipping the number.It's important to remember that:

• The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).
• The reciprocal of an integer \( n \) is \( \frac{1}{n} \).

For negative numbers like \(-10\), the reciprocal, or multiplicative inverse, becomes \(-\frac{1}{10}\).Reciprocals are particularly useful in complex algebra problems and whenever you need to divide numbers in math.

Algebra Concepts

The concepts of additive and multiplicative inverses are part of fundamental algebra principles. Understanding these can help you navigate through more complex algebraic problems. Additive inverse is used to balance equations, while multiplicative inverse simplifies division and fraction operations. Both concepts help achieve simpler forms in equations so you can solve them with greater ease. Knowing how to find and apply these inverses allows:

• Efficient problem-solving in algebra.
• Simplification of complex expressions.
• Better comprehension of mathematical structures and theories.

Mastering these inverses builds a strong foundation for further mathematical learning and application.