Problem 7 - College Algebra with Modeling and Visualization

Question

Describe verbally the inverse of the statement. Then express both the given statement and its inverse symbolically. Subtract 2 from \(x\) and multiply the result by 3 .

Step-by-Step Solution

Answer

Inverse: Divide by 3, then add 2; Symbolic Inverse: \( x = \frac{y}{3} + 2 \).

Step 1: Understand the Given Statement

The original statement asks us to subtract 2 from a variable, which we can denote as \( x \), and then multiply the result by 3. Symbolically, this can be written as \( 3(x - 2) \).

Step 2: Understand Inverse Operation

To find the inverse of a statement involving mathematical operations, we perform the opposite operations in the reverse order. The given statement describes first subtracting 2 and then multiplying by 3. The inverse would start by dividing by 3 and then adding 2.

Step 3: Develop Verbal Inverse Statement

The inverse of the statement 'subtract 2 from \( x \) and multiply the result by 3' is 'divide by 3 and then add 2 to the result'. This correctly reverses and applies opposite operations.

Step 4: Symbolic Representation of the Inverse

If we have \( y = 3(x - 2) \), the inverse operation aims to solve for \( x \) in terms of \( y \). Begin by dividing both sides of the equation by 3: \( y/3 = x - 2 \). Then, add 2 to both sides to isolate \( x \): \( x = y/3 + 2 \). Thus, the inverse in symbolic form is \( x = \frac{y}{3} + 2 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are a way of writing mathematical phrases using numbers, variables, and arithmetic operations. In the given exercise, we worked with the expression \( 3(x - 2) \). This combines a number, 3, a variable, \( x \), and the operation of subtraction and multiplication.

An algebraic expression like this one allows us to perform calculations universally with any value that \( x \) might take. It follows a systematic structure which makes it quicker and easier to work with equations.

In our exercise, understanding the algebraic expression was crucial for both representing the original statement symbolically and in identifying the inverse operation. By understanding that expressions like \( 3(x - 2) \) have specific operations in sequence, students can manipulate and solve these equations systematically.

Symbolic Representation

Symbolic representation is simply expressing verbal statements using symbols and mathematical notation. It provides a convenient and compact way to visualize and work through mathematical concepts.

In our exercise, the given statement 'subtract 2 from \( x \) and multiply by 3' was represented symbolically as \( 3(x - 2) \). Each word in the sentence corresponded to mathematical symbols: subtraction was represented as \( -2 \) and multiplication as \(*)3\).

Similarly, for the inverse statement, 'divide by 3 and then add 2,' we derived the symbolic expression \( x = \frac{y}{3} + 2 \). This inverse expression symbolizes performing the exact opposite and reverse operations. Thus, learning symbolic representation enables us to deal with complex equations more effectively.

Verbal Description

Verbal description involves converting mathematical operations into plain language explanations. It helps in understanding and communicating math concepts effectively without always relying on symbols.

The exercise required describing the inverse of the statement in words. Originally stated as 'subtract 2 from \( x \) and multiply the result by 3,' its verbal inverse is 'divide by 3 and then add 2 to the result.'

By accurately using verbal descriptions, one makes math more accessible, especially when explaining processes or understanding problems intuitively before writing them in symbolic form. This skill ensures that complex operations are broken down into basic steps that can be communicated and understood easily.