Problem 92 - 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. $$ \text { Opposite: } $$ $$ 7 x $$

Step-by-Step Solution

Answer

The opposite of \(7x\) is \(-7x\).

Step 1: Identify the Opposite

The opposite or additive inverse of any number or expression is simply the negative of that expression. Thus, for an expression to find its opposite, we need to multiply it by (-1). For the expression given as \(7x\), its opposite would be \(-7x\).

Step 2: Understand Additive Inverses

Recall that if you have a number \(a\), its additive inverse is \(-a\). When you add them together, \(a + (-a) = 0\). This means they cancel out each other. So, the additive inverse of \(7x\) being \(-7x\) fits into this definition.

Step 3: Write the Solution

Given the expression \(7x\), the solution for the opposite is \(-7x\). Thus, we fill in the table with the value \(-7x\) under the opposite category.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers and variables linked by mathematical operations such as addition, subtraction, multiplication, and division. Each part of the expression separated by a plus or minus sign is called a "term". For example, in the expression \(7x\), "7" is a coefficient, "x" is a variable, and together they form a single term.
Understanding these expressions is essential as they form the basis of algebra. With variables representing unknowns or quantities that can change, algebraic expressions allow us to describe and solve real-world problems.
You might encounter expressions with multiple terms, like

• \(3x + 4y - 5\)
• \(x^2 + 2x + 1\)

Grasping how these expressions work sets the stage for solving equations, working with functions, and more complex algebra.

Multiplicative Inverse

The multiplicative inverse, or reciprocal, of a number is what you multiply that number by to get a product of 1. In simple terms, for a number \(a\), its multiplicative inverse is \(\frac{1}{a}\). This concept is vital in division and solving equations.
For example, if you have the number 5, its multiplicative inverse is \(\frac{1}{5}\), because \(5 \times \frac{1}{5} = 1\). The same holds for more complex expressions such as \(7x\). Here, the multiplicative inverse is \(\frac{1}{7x}\), assuming \(x eq 0\).
Understanding multiplicative inverses is key to simplifying complex fractions and solving algebraic equations efficiently, as they help "cancel" terms to isolate the variable or simplify the equation.

Mathematical Operations

Mathematical operations are foundational processes used in mathematics, including addition, subtraction, multiplication, and division. They serve as the building blocks for solving equations and manipulating algebraic expressions.
Each operation has specific rules, such as the order of operations (often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)).
In algebra, these operations are applied to variables and numbers alike, allowing us to transform and simplify expressions and equations. For instance:

• In addition, \(x + 3x = 4x\).
• For multiplication, \(2x \times 3 = 6x\).
• In division, dividing \(6x^2\) by 3x results in \(2x\).

Mastering these operations ensures you can handle more complex mathematical tasks and effectively work through various algebraic problems.