Problem 92 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The difference between 9 times a number and \(-4\) times the number

Step-by-Step Solution

Answer

The simplified algebraic expression is \(13x\).

Step 1: Translate English phrase into algebraic expression

The English phrase 'The difference between 9 times a number and -4 times the number' can be translated into an algebraic expression as follows: This means \(9x - (-4x)\), where 'x' represents the number.

Step 2: Simplify the algebraic expression

The expression \(9x - (-4x)\) simplifies to \(9x + 4x\). This is because subtracting a negative number is the same as adding the positive number.

Step 3: Add like terms

In terms of \(x\), the expression \(9x + 4x\) simplifies to \(13x\). This is done because \(9x\) and \(4x\) are like terms, and when they are added, we get \(13x\).

Key Concepts

Translating Phrases into Expressions

When translating English phrases into algebraic expressions, the key is identifying the mathematical operations described by words. In our given exercise, we need to express the phrase "the difference between 9 times a number and -4 times the number." Here's how to approach this:

• "9 times a number" is represented by the expression \(9x\).
• "-4 times the number" is represented by \(-4x\).
• "The difference between" indicates subtraction.

Thus, when these elements are combined, the expression becomes \(9x - (-4x)\), where \(x\) stands for the unknown number. This process involves recognizing operations such as addition, subtraction, and multiplication, and their corresponding phrases. Breaking down and understanding each part of the phrase makes translating it into an expression straightforward.

Simplifying Expressions

Once you have translated the phrase into an algebraic expression, the next step is simplification. Simplification helps make expressions easier to work with and understand. In mathematics, simplifying means combining or reducing the parts of an expression to a simpler form.
If we look at \(9x - (-4x)\), note that subtracting a negative is the same as adding its positive counterpart. So,

• We transform \(9x - (-4x)\) into \(9x + 4x\).
• This action simplifies the expression by removing the subtraction of a negative.

This process shows how simplifying can lead to a clearer and simplified representation of the original problem, avoiding any potential confusion related to the myriad of operations involved.

Like Terms

When working with algebraic expressions, understanding like terms is crucial for simplifying expressions further. Like terms are parts of an expression that have the same variables raised to the same power. They can be combined through addition or subtraction.
In the expression \(9x + 4x\), both terms, \(9x\) and \(4x\), are like terms because they have the same variable \(x\). This similarity allows us to combine them:

• We add the coefficients (numbers in front of \(x\)): \(9 + 4 = 13\).
• The expression simplifies to \(13x\).

By recognizing and combining like terms, you can simplify expressions to their most reduced form. This skill is fundamental in algebra as it helps us see the relationships between terms more clearly.