Problem 128 - College Algebra

Question

Write 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 written as the algebraic expression \(-3x -10\).

Step 1: Interpret the English phrase

The phrase 'Eight decreased by three times the sum of a number and six' includes several parts. It first mentions 'eight decreased by', which indicates a subtraction operation. The 'three times' represents multiplication. Lastly, 'the sum of a number and six' indicates an addition operation. But importantly, the operations should be done in the correct order according to the basic mathematical order of operations (PEMDAS/BODMAS)

Step 2: Write the algebraic expression

Converting the phrase to an algebraic expression using variable \(x\), the expression becomes:'Eight decreased by three times the sum of a number and six' -> \(8 - 3(x + 6)\)

Step 3: Simplify the algebraic expression

With basic algebraic simplification, distribute the '-3' inside the bracket: \(8 - 3x - 18\). With further simplification by adding constants gives: \(-3x -10\).

Key Concepts

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. This makes it easier to understand and work with in calculations. When simplifying expressions, you look for ways to combine like terms and eliminate unnecessary parts.
For instance, in the given expression, we start with \(8 - 3(x + 6)\). The key step is to use the distributive property to unfold the terms in the brackets. Once you distribute the -3 across each term inside the parentheses, you get \(8 - 3x - 18\).
Then, move on to combine like terms. In this case, you have constants (8 and -18) which can be combined to form \(-10\). This gives the simplified expression: \(-3x - 10\).
Simplification helps in revealing the essence of the expression and is crucial for solving algebraic equations effectively.

Order of Operations

The order of operations is essential in algebra to ensure expressions are simplified correctly, especially when they involve multiple arithmetic operations such as addition, subtraction, multiplication, division, and parentheses. This sequence is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
For the expression \(8 - 3(x + 6)\), applying the order of operations means starting with the operations inside the parentheses.

• Calculate the expression within the parentheses first: \(x + 6\).
• Next, handle the multiplication: \(-3(x + 6)\), giving \(-3x - 18\).
• Finally, perform the subtraction: \(8 - 3x - 18\), which simplifies further to \(-3x - 10\).

Understanding and applying the order of operations correctly ensures that algebraic expressions are handled in a consistent and error-free manner.

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a sum by distributing the multiplication over each addend within parentheses. This property is expressed as \(a(b + c) = ab + ac\).
In our example, the distributive property is used to simplify the expression \(8 - 3(x + 6)\). Here, \(-3\) multiplies both \(x\) and \(6\), resulting in \(-3x - 18\).
This property ensures that multiplication is carried out accurately across terms within parentheses, providing a path to simplify even more complex algebraic expressions. Understanding how to apply the distributive property enables students to break down and solve equations that might otherwise seem cumbersome.
Mastering this technique is crucial for progressing in more advanced algebra and ensuring clarity in mathematical reasoning.