Problem 82 - Algebra 2

Question

PREREQUISITE SKILL Identify the additive inverse for each number or expression. \(-3 x\)

Step-by-Step Solution

Answer

The additive inverse of \(-3x\) is \(3x\).

Step 1: Understanding Additive Inverse

The additive inverse of a number or expression is a value that, when added to the original number or expression, results in zero. In mathematical terms, the additive inverse of a number \(a\) is \(-a\). The goal is to find a value that will cancel out the original expression.

Step 2: Applying the Concept

Given the expression \(-3x\), identify its additive inverse by changing its sign. Since \(-3x\) has a negative sign, its additive inverse will be positive. Thus, the additive inverse of \(-3x\) is \(3x\).

Step 3: Verification

To verify, add the original expression \(-3x\) to its additive inverse \(3x\). The sum is \(-3x + 3x = 0\), which confirms that \(3x\) is indeed the correct additive inverse of \(-3x\).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations such as adding, subtracting, multiplying, and dividing. In algebraic expressions, variables represent unknown values that can be solved for or expressed in various forms. For example, in the expression \(-3x\), the \(-3\) is a coefficient, and \(x\) is the variable. Keep in mind:

• Coefficients are numbers multiplying the variables in an expression.
• Variables are symbols that stand in for unknown numbers.
• Operations can be addition, subtraction, multiplication, or division.

Algebra helps us to solve equations and find values for unknowns by manipulating these expressions. In our exercise, identifying the additive inverse of \(-3x\) seems simple, but it hinges on understanding how expressions are built and altered. By switching the sign, as we do when finding additive inverses, we change the intrinsic value of the expression, helping to balance equations.

Negative Numbers

Negative numbers are numbers less than zero. They are important in everyday math, especially in topics like temperature, debt, and elevation, where values can move below zero. Understanding negative numbers is crucial when working with expressions and equations.

They work under some basic rules:

• Adding a negative number is the same as subtracting its positive counterpart. For example, \(5 + (-2) = 3\).
• Multiplying two negative numbers gives a positive result (for example, \((-2) \times (-3) = 6\)).
• Dividing a number by a negative makes the quotient negative if the dividend is positive.
• Adding the opposite of a negative number results in zero, showcasing the principle of additive inverses.

When we look at the expression \(-3x\), the \(-3\) indicates a negative coefficient. By finding its additive inverse, \(3x\), we effectively neutralize the original expression when added together. This action highlights one purpose of understanding negative numbers - achieving balance and zeroing out expressions.

Mathematical Concepts

Mathematical concepts form the foundation for solving a variety of problems. Familiarity with these basic principles enables one to understand more complex ideas over time. One key concept is the additive inverse, which helps to simplify expressions and solve equations.

Additive inverse means you find a number which, when added to the original number, equals zero. This principle can also be referred to as 'canceling' or 'zeroing out' an expression.

• For any number \(a\), \(-a\) is its additive inverse.
• This principle helps solve for unknowns by simplifying and reducing equations.
• Understanding the additive inverse aids in equation solving techniques such as isolation of variables.

In the given exercise, solving for the additive inverse of \(-3x\) provides insights into how algebra represents balancing equations. Such concepts are not only powerful in equations but help in comprehending broader mathematical logic and applications.