Problem 87 - Beginning and Intermediate Algebra

Question

Fill in the table with the opposite (additive inverse), and the reciprocal (multiplicative inverse). Assume that the value of each expression is not 0 $$x$$

Step-by-Step Solution

Answer

Additive inverse of \(x\) is \(-x\); multiplicative inverse is \(\frac{1}{x}\) if \(x \neq 0\).

Step 1: Understand Additive Inverse

The additive inverse of a number is what you add to a number to get zero. For any number \(x\), its additive inverse is \(-x\).

Step 2: Identify the Additive Inverse

Given the number \(x\), its additive inverse is \(-x\). This fulfills the condition: \(x + (-x) = 0\).

Step 3: Understand Multiplicative Inverse

The multiplicative inverse of a number is what you multiply a number by to get 1. For any number \(x\), assuming \(x eq 0\), its multiplicative inverse is \(\frac{1}{x}\).

Step 4: Identify the Multiplicative Inverse

Given the number \(x\), its multiplicative inverse is \(\frac{1}{x}\). This fulfills the condition: \(x \times \frac{1}{x} = 1\).

Step 5: Fill in the Table

For the given number \(x\), fill in the table with the additive inverse as \(-x\) and the multiplicative inverse as \(\frac{1}{x}\) assuming \(x eq 0\).

Key Concepts

Additive Inverse

The additive inverse is a fundamental concept in algebra. When you hear 'additive inverse,' think of the number that, when added to the original, results in zero. For any real number \( x \), the additive inverse is simply \( -x \). This means if you start with a number and its additive inverse, their sum will always be zero:
\[ x + (-x) = 0 \]

• The additive inverse is a straightforward concept because it's about finding the 'opposite number' in terms of addition.
• For positive numbers, the additive inverse will be negative. For instance, the additive inverse of 5 is -5.
• Conversely, the additive inverse of a negative number flips it to positive. So, the additive inverse of -4 is 4.
• Zero is unique, as its additive inverse is itself, since \( 0 + 0 = 0 \).

Understanding and identifying the additive inverse helps in solving equations and simplifying expressions by neutralizing values.

Multiplicative Inverse

The multiplicative inverse, or reciprocal, is the number that, when multiplied with the original number \( x \), gives you one. For non-zero numbers, the multiplicative inverse of \( x \) is \( \frac{1}{x} \). This inverse should not be confused with division; it is the exact value you multiply to achieve the identity, one:
\[ x \times \frac{1}{x} = 1 \]

• If \( x = 1 \), its multiplicative inverse remains \( 1 \), because \( 1 \times 1 = 1 \).
• For a fraction, its multiplicative inverse is simply flipping the numerator and denominator. For example, the inverse of \( \frac{3}{4} \) is \( \frac{4}{3} \).
• Negative numbers possess a similar rule; the reciprocal of \( -x \) is \( \frac{-1}{x} \), since \( -x \times \frac{-1}{x} = 1 \).
• Zero is unique because it doesn’t have a multiplicative inverse. This is because any number multiplied by zero is still zero, and it never equals one.

Recognizing the multiplicative inverse is crucial for solving fractional equations and simplifying complex algebraic expressions.

Inverse Operations

Inverse operations are pairs of mathematical processes that reverse each other. They are especially useful in algebra to solve equations and simplify computations.
Inverse operations include:

• Addition and Subtraction: These operations are opposites. Adding a number and then subtracting the same number naturally cancels them out. For example, if you add 7 and then subtract 7, you end up where you started.
• Multiplication and Division: Multiplying and then dividing by the same number leaves you with your original number. For instance, multiplying by 2 and then dividing by 2 returns you to your starting value.

Understanding inverse operations enables quicker problem-solving because it equips you with strategies to cancel terms or factors.

• They allow you to isolate variables or simplify expressions by applying the opposite operation.
• They help in breaking down equations in manageable steps by removing unwarranted complexity.
• Keeps equations balanced by performing the same operation on both sides to maintain equality.

In essence, using inverse operations is a robust strategy to tackle mathematical problems efficiently and accurately.