Problem 88 - Algebra A Combined Function

Question

Fill in the table with the opposite (additive inverse), the reciprocal (multiplicative inverse), or the expression. Assume that the value of each expression is not 0. $$ \begin{array}{|c|c|c|} \hline & \text { 88. } \\ \hline \text { Expression } & -\frac{2}{3} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Additive inverse: \(\frac{2}{3}\); Multiplicative inverse: \(-\frac{3}{2}\).

Step 1: Understanding the Expression

The given expression is \(-\frac{2}{3}\). We need to find both the additive inverse and the multiplicative inverse of this fraction.

Step 2: Finding the Additive Inverse

The additive inverse (opposite) of a number \(x\) is the number that, when added to \(x\), equals zero. For the expression \(-\frac{2}{3}\), its additive inverse is simply \(\frac{2}{3}\), because \(-\frac{2}{3} + \frac{2}{3} = 0\).

Step 3: Finding the Multiplicative Inverse

The multiplicative inverse (reciprocal) of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). For \(-\frac{2}{3}\), the reciprocal is \(-\frac{3}{2}\), which satisfies the condition that \(-\frac{2}{3} \times -\frac{3}{2} = 1\).

Step 4: Completing the Table

Based on the steps, we can fill the table: the additive inverse of \(-\frac{2}{3}\) is \(\frac{2}{3}\), and the multiplicative inverse is \(-\frac{3}{2}\).

Key Concepts

Additive Inverse

Every number has an **additive inverse**. This is the number that, when combined with the original number, results in zero. Think of it as finding the 'opposite' number. For any given number \( x \), its additive inverse is \( -x \). It's quite straightforward: just change the sign of the number!

For example, the additive inverse of \(-\frac{2}{3}\) is \(\frac{2}{3}\) because adding these two numbers gives you zero:

• \(-\frac{2}{3} + \frac{2}{3} = 0\)

This concept is useful because it helps us understand operations that bring us back to 'balance', or zero, in mathematical equations.

This property is especially helpful in solving equations where getting rid of a term can simplify the expression. In practice, when solving for a variable, using the additive inverse allows terms to be canceled out.

Multiplicative Inverse

The **multiplicative inverse**, or **reciprocal**, of a number is the value that, when multiplied by the original number, equals one. This is an essential concept, particularly when dealing with division and fractions.

For a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). However, if a fraction is negative, the reciprocal will also be negative to maintain the requirement that the product equals one.

• For \(-\frac{2}{3}\), the multiplicative inverse is \(-\frac{3}{2}\).
• Checking: \(-\frac{2}{3} \times -\frac{3}{2} = 1\)

This inverse is pivotal when solving equations that require division. Multiplying by the reciprocal can effectively "undo" multiplication.It's equivalent to dividing by a number, which is why knowing how to find multiplicative inverses is useful and efficient.

Fraction Operations

Working with fractions involves understanding both the **additive** and **multiplicative inverses**. Fractions represent a part of a whole and are written in the format \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) the denominator.
When finding operations involving fractions:

• **Addition/Subtraction** requires a common denominator.
• **Multiplication** is straightforward: multiply numerators together and multiply denominators together.
• **Division** is essentially the same as multiplying by the reciprocal of the fraction.

Understanding these helps us manipulate fractions, making them easier to work with in mathematical problems and real-life applications. Knowing the inverses of fractions can simplify these operations, particularly in solving algebraic equations where fractions appear.