Problem 90 - 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 $$ 7 x $$

Step-by-Step Solution

Answer

Opposite: \( -7x \); Reciprocal: \( \frac{1}{7x} \).

Step 1: Understanding the Additive Inverse

The additive inverse of a number is the number that, when added to the original number, yields zero. For any number \( a \), its additive inverse is \( -a \). Therefore, the additive inverse of \( 7x \) is \( -7x \).

Step 2: Understanding the Multiplicative Inverse

The multiplicative inverse (or reciprocal) of a non-zero number is the number that, when multiplied by the original number, yields one. For any number \( a \), its reciprocal is \( \frac{1}{a} \). Thus, the reciprocal of \( 7x \) is \( \frac{1}{7x} \) given that \( 7x eq 0 \).

Step 3: Filling in the Table

Now, with the additive inverse \( -7x \) and the reciprocal \( \frac{1}{7x} \) calculated, fill in the table with these values: - Opposite (Additive Inverse) of \( 7x \): \( -7x \) - Reciprocal (Multiplicative Inverse) of \( 7x \): \( \frac{1}{7x} \).

Key Concepts

Additive Inverse

In algebra, the concept of an additive inverse is crucial when performing operations and solving equations. The additive inverse of a number is the value that will make the sum equal to zero when added to the original number. Think of it as the "opposite" of the number.

To illustrate, consider the number 5. The additive inverse of 5 is -5 because:

• Adding 5 and -5 results in 0. In equation form: \( 5 + (-5) = 0 \).
• This works for any number, including algebraic expressions like \(7x\). Here, the additive inverse is \(-7x\), since \(7x + (-7x) = 0\).

Understanding this helps in simplifying expressions and solving equations, especially those requiring terms to be moved from one side of an equation to the other.

Multiplicative Inverse

The multiplicative inverse, commonly known as the reciprocal, plays a vital role in division and fraction operations. The multiplicative inverse of a number is what you multiply the number by to get a product of one.

For example, the multiplicative inverse of 8 is \( \frac{1}{8} \), because:

• Multiplying 8 by its reciprocal gets you 1. In equation form: \( 8 \times \frac{1}{8} = 1 \).
• Listeners should note that this concept applies to any algebraic expression. For \(7x\), the reciprocal is \(\frac{1}{7x}\). Multiplying \(7x\) by \(\frac{1}{7x}\) results in 1: \( 7x \times \frac{1}{7x} = 1 \) provided \(7x eq 0\).

This is fundamental when dividing by fractions or rearranging equations, simplifying many algebraic tasks.

Reciprocal

Reciprocal may sound like a complex term, but it's quite straightforward. A reciprocal is essentially the flipped version of a number, especially focusing on fractions. If you take any number \(a\), its reciprocal is \(\frac{1}{a}\).

Here's how it works:

• For the number 4, the reciprocal is \( \frac{1}{4} \), since \( 4 \times \frac{1}{4} = 1 \).
• Every non-zero number has a reciprocal, and for algebraic expressions like \(7x\), its reciprocal is \(\frac{1}{7x}\), because multiplying \(7x\) by \(\frac{1}{7x}\) equals 1.

Reciprocals are often used to solve equations or divide fractions by essentially multiplying by the reciprocal, making them a powerful tool in algebra.