Problem 32 - Algebra 2

Question

Identify the additive inverse and multiplicative inverse for each number. $$ \frac{4}{3} $$

Step-by-Step Solution

Answer

Additive inverse: \(-\frac{4}{3}\); Multiplicative inverse: \(\frac{3}{4}\).

Step 1: Define the Problem

We need to find both the additive inverse and the multiplicative inverse of the number \( \frac{4}{3} \). The additive inverse is the number that, when added to the original number, results in zero. The multiplicative inverse is the number that, when multiplied by the original number, results in one.

Step 2: Find the Additive Inverse

The additive inverse of a number \( a \) is \( -a \). For the number \( \frac{4}{3} \), the additive inverse is \( -\frac{4}{3} \) because \( \frac{4}{3} + (-\frac{4}{3}) = 0 \).

Step 3: Find the Multiplicative Inverse

The multiplicative inverse of a number \( a \) is \( \frac{1}{a} \). For the number \( \frac{4}{3} \), the multiplicative inverse is \( \frac{3}{4} \) because \( \frac{4}{3} \times \frac{3}{4} = 1 \).

Key Concepts

Additive Inverse

When dealing with inverse operations, the additive inverse is an essential concept. Simply put, the additive inverse of a number is what you add to it to get zero. This means if you have a number, say 'a', its additive inverse is '-a'.
To make this clear, let's take the fraction \( \frac{4}{3} \) as an example. The additive inverse of \( \frac{4}{3} \) would be \(-\frac{4}{3}\). Why? Because when you add \( \frac{4}{3} \) and \(-\frac{4}{3}\), they cancel each other out, leading to a sum of zero:
\[ \frac{4}{3} + (-\frac{4}{3}) = 0 \]
This property is crucial in solving equations as it's often used to isolate variables by making terms disappear. Remember:

• Additive inverse flips the sign.
• Sum of a number and its additive inverse is always zero.

Multiplicative Inverse

The multiplicative inverse, often called the reciprocal, is what you multiply a number by to get one. For any non-zero number 'a', its multiplicative inverse is \( \frac{1}{a} \). This means if you multiply 'a' and \( \frac{1}{a} \), the result is always one.
Taking \( \frac{4}{3} \) as our number, its multiplicative inverse is \( \frac{3}{4} \). See how it works:
\[ \frac{4}{3} \times \frac{3}{4} = 1 \]
It's as if the two numbers "undo" each other by multiplication, always leading back to the concept of "one". Here are the key points:

• Multiplication of a number by its inverse yields one.
• For \( a \), the multiplicative inverse is \( \frac{1}{a} \).
• Both numbers must be non-zero.

Fractions

Fractions might initially seem complex, but they are simply a way of representing parts of a whole. A fraction has two parts: the numerator (top number) and the denominator (bottom number).
In \( \frac{4}{3} \), 4 is the numerator, and 3 is the denominator. This fraction is "improper" because the numerator is larger than the denominator, indicating a value greater than one.
Working with fractions involves several operations, such as addition, subtraction, multiplication, and division. A crucial skill is simplifying fractions, where you reduce them to their smallest form by dividing both the numerator and denominator by their greatest common divisor (GCD).
Additionally, understanding how to find the inverse also involves fraction skills:

• Swapping the numerator and denominator gives the reciprocal.
• Sign flipping helps in identifying additive inverses.

Fractions also play a key role in real-life scenarios, such as dividing a pizza or measuring ingredients. With practice, they become a straightforward way to understand parts and wholes.