Problem 88 - 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 $$ 4 y $$

Step-by-Step Solution

Answer

Opposite: \(-4y\); Reciprocal: \(\frac{1}{4y}\).

Step 1: Identify the Given Expression

We start with the expression given: \(4y\). Our task is to find both its opposite (additive inverse) and its reciprocal (multiplicative inverse).

Step 2: Find the Opposite (Additive Inverse)

The opposite of a number is obtained by changing its sign. Therefore, the opposite of \(4y\) is \(-4y\).

Step 3: Find the Reciprocal (Multiplicative Inverse)

To find the reciprocal of an expression, we take the inverse of it by flipping the numerator and the denominator. Therefore, the reciprocal of \(4y\) is \(\frac{1}{4y}\).

Key Concepts

Additive Inverse

The additive inverse of a number or expression is what you add to that number to get a sum of zero. It’s like the opposite of the number on the number line. For instance, the additive inverse of \(4y\) is \(-4y\). Here’s how to find the additive inverse:

• Take the original number or expression
• Change its sign

So, if you have \(4y\), its additive inverse is \(-4y\) because \(4y + (-4y) = 0\).
This concept helps in solving equations and simplifying problems. Whenever you see an equation in the form of \(a + b = 0\), remember that \(b\) is the additive inverse of \(a\).
It's a simple yet powerful tool in algebra that ensures symmetry and balance in equations.

Multiplicative Inverse

The multiplicative inverse, also known as the reciprocal, is a number which when multiplied with the original number, produces the product of one. This is incredibly useful in division problems and solving fractions.

To find the multiplicative inverse:

• If you have a fraction, just flip the numerator and the denominator.
• If it's a whole number or variable expression, place it under one, making it a fraction.

For example, if the expression is \(4y\), its multiplicative inverse is \(\frac{1}{4y}\).
This is because when you multiply \(4y\) by \(\frac{1}{4y}\), the result is 1:\[4y \times \frac{1}{4y} = 1\]The concept of the multiplicative inverse is vital for understanding division and can simplify many algebraic fractions or rational expressions. Remember, the goal with multiplicative inverses is always achieving the value of one.

Reciprocal

The term 'reciprocal' is often interchangeable with 'multiplicative inverse', which makes understanding its simplicity easier.
When you look for the reciprocal of any given expression or number, you're essentially looking for what you must multiply it by to get the result of 1.

Let’s break it down:

• For a number like 5, the reciprocal is \(\frac{1}{5}\).
• For a variable such as \(x\), the reciprocal is \(\frac{1}{x}\).
• For an expression like \(4y\), the reciprocal is \(\frac{1}{4y}\).

This simple "flip" of the number or expression allows for easier calculations, especially when dealing with division and fractions.
Moreover, understanding reciprocals is crucial in solving complex algebraic problems, as it facilitates getting isolated variables and simplifying expressions.