Problem 13 - Beginning and Intermediate Algebra

Question

Complete the table and each statement beside it. $$\begin{array}{|r|l|l|}\hline \text { Number } & \text { Additive Inverse } & \text { Multiplicative Inverse } \\\\\hline 5 & & \\\\\hline-10 & & \\\\\hline-\frac{1}{2} & & \\\\\hline \frac{3}{8} & & \\\\\hline x & & (x \neq 0)\\\\\hline-y & & (y \neq 0)\\\ \hline \end{array}$$ A number and its additive inverse have (the same/opposite) signs. A number and its multiplicative inverse have (the same / opposite) signs.

Step-by-Step Solution

Answer

Add the additive and multiplicative inverse to the table as shown. Numbers and their additive inverse have opposite signs, while numbers and their multiplicative inverse can have the same or opposite signs.

Step 1: Understand Additive Inverse

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

Step 2: Understand Multiplicative Inverse

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

Step 3: Complete the Table

Using the definitions of additive and multiplicative inverses, complete the table as follows:\( \begin{array}{|r|l|l|}\hline\text{Number} & \text{Additive Inverse} & \text{Multiplicative Inverse} \hline5 & -5 & \frac{1}{5} \hline-10 & 10 & -\frac{1}{10} \hline-\frac{1}{2} & \frac{1}{2} & -2 \hline\frac{3}{8} & -\frac{3}{8} & \frac{8}{3} \hlinex & -x & \frac{1}{x} \,\text{(where } x eq 0\text{)} \hline-y & y & -\frac{1}{y} \,\text{(where } y eq 0\text{)} \hline\end{array} \)

Step 4: Statements Analysis

To determine the statements:- A number and its additive inverse have opposite signs. For example, \(5\) and \(-5\) have opposite signs.- A number and its multiplicative inverse can have either the same or opposite signs depending on the number. For positive numbers, the multiplicative inverse is positive, and for negative numbers, the multiplicative inverse is negative. For example, \(5\) and \(\frac{1}{5}\) have the same sign, while \(-10\) and \(-\frac{1}{10}\) have the same sign.

Key Concepts

Additive Inverse

The concept of additive inverse is quite straightforward. Think about it this way: To find an additive inverse of any number, ask yourself what you need to add to that number to get zero. For any number \(a\), its additive inverse is \(-a\). This means if you start with 5, its additive inverse is -5, because 5 + (-5) equals zero. Easy, right? Here are some useful points to remember:
- The additive inverse of \(a\) is always \(-a\).
- Adding a number to its additive inverse always results in zero.
- This helps in balancing equations and solving algebraic problems.

Multiplicative Inverse

Multiplicative inverse, also known as reciprocal, is another fundamental concept in algebra. To find the multiplicative inverse of a number, you're looking for a number that, when multiplied by the original number, results in one. For any non-zero number \(a\), the multiplicative inverse is \(\frac{1}{a}\). Let's check out how this works:
- For example, the multiplicative inverse of 5 is \(\frac{1}{5}\) because 5 * \(\frac{1}{5}\) = 1.
- Similarly, the multiplicative inverse of -10 is \(-\frac{1}{10}\) because -10 * \(-\frac{1}{10}\) = 1.
- Note that zero has no multiplicative inverse because you cannot divide by zero.

Opposite Signs

Understanding the signs of numbers and their inverses is crucial. Let's delve into this:
- **Additive Inverses**: A number and its additive inverse will always have opposite signs. For instance, if the number is positive, its additive inverse is negative, and vice versa. Let's consider 3. Its additive inverse is -3. Clearly, they have opposite signs.
- **Multiplicative Inverses**: The signs of the number and its multiplicative inverse can vary. If your number is positive, such as 5, both the number and its reciprocal (\(\frac{1}{5}\)) are positive. If your number is negative, like -10, then both the number and its reciprocal (\(-\frac{1}{10}\)) are negative too.
- This understanding helps in many algebraic operations and simplifies your calculations.