Problem 128 - College Algebra Essentials

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. Eight decreased by three times the sum of a number and six.

Step-by-Step Solution

Answer

The English phrase 'Eight decreased by three times the sum of a number and six' can be expressed and simplified into the algebraic expression \(-3x - 10\).

Step 1: Identifying the operations

Identify the operations contained within the phrase. In this case, we have identified subtraction (from 'decreased by'), addition (from 'sum'), and multiplication (from 'times').

Step 2: Formulate the algebraic expression

Translate the phrase to the algebraic expression. 'Eight decreased by' denotes 8 - something, and 'three times the sum of a number and six' denotes 3 * (x + 6). Combining these gives us the expression \(8 - 3 \times (x + 6)\).

Step 3: Simplify the expression

The last step is to simplify the algebraic expression. Distribute the 3 in \(3 \times (x + 6)\), then do the subtraction with 8 which results in \(8 - 3x - 18\). Simplifying this further gives the final expression \(-3x - 10\).

Key Concepts

Operations in Algebra

Algebra involves different mathematical operations, such as addition, subtraction, multiplication, and division. Understanding these operations is essential when dealing with algebraic expressions. Identifying what each word or phrase means in terms of mathematical operations is the first step in translating a written phrase into an algebraic expression.

• **Subtraction** is often indicated by phrases like "decreased by" or "minus."
• **Addition** might be represented by words like "plus" or "sum."
• **Multiplication** can be indicated by "times" or "product."
• **Division** might be expressed through words such as "divided by" or "quotient."

In our example, the phrase "eight decreased by three times the sum of a number and six" involves subtraction (decreased by), multiplication (times), and addition (sum). Recognizing these operations helps in correctly formulating the corresponding algebraic expression.

Translating Phrases to Expressions

Converting an English phrase into an algebraic expression involves analyzing the phrase and assigning mathematical symbols to words or groups of words. This translation requires precision and understanding of what each component of the phrase is asking you to do.
Here's how it works in our example:

• "Eight decreased by" indicates that 8 is our starting point and we will be subtracting from it.
• "Three times the sum of a number and six" tells us we need to first find the sum of a number (represented by **x**) and 6, then multiply that sum by 3.
• Combining these ideas in algebraic language results in the expression: \[8 - 3 \times (x + 6)\]

This step ensures that the algebraic expression precisely represents the verbal description of the situation.

Simplifying Algebraic Expressions

After forming the algebraic expression, the next step is to simplify it to make it as straightforward as possible. Simplification involves combining like terms and performing any arithmetic operations that can make the expression simpler.
In our example, the expression is \(8 - 3 \times (x + 6)\). Simplifying involves:

• Start by distributing the 3 into the parentheses: \[3 \times (x + 6) = 3x + 18\]
• Plug this result back into the original expression: \[8 - 3x - 18\]
• Combine like terms, which in this case, are the constant numbers 8 and -18: \[8 - 18 = -10\]
• The fully simplified expression is \[-3x - 10\]

Simplifying not only makes the algebraic expression easier to understand but also prepares it for use in solving equations or further operations.