Problem 66 - Elementary Algebra Within Reach

Question

In Exercises 63-68, translate the verbal phrase into an algebraic expression. Simplify the expression. $$ \text { The sum of } 4 \text { and } x \text { added to the sum of } x \text { and }-8 $$

Step-by-Step Solution

Answer

The simplified algebraic expression for the given verbal phrase is \(2x - 4\).

Step 1: Translate Verbal Phrases into Algebraic Expressions

The verbal phrase 'the sum of 4 and x' can be translated into the algebraic expression '4 + x'. Similarly, the verbal phrase 'the sum of x and -8' can be translated into 'x - 8'.

Step 2: Use the word 'added' to form the complete expression

The word 'added' indicates that the results of the two sums should be added together. This gives the complete algebraic expression '4 + x + x - 8'.

Step 3: Simplify the Expression

Now, we simplify the expression by combining like terms. This gives '4 + 2x - 8'.

Step 4: Final Simplification

Finally, combine the constants to get the simplified expression '2x - 4'.

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their most basic form. It means transforming a complex expression into a simpler one while maintaining its value. This can often make computations easier and provide clearer insights into the nature of the expression.
When simplifying, follow these key steps:

• Combine like terms: As in our example, an expression such as '4 + x + x - 8' can be simplified by adding similar terms together to make it easier to handle.
• Conduct arithmetic operations: Sum or subtract constants. For example, '4 - 8' simplifies to '-4', making the expression '2x - 4'.

Simplification is a critical skill in algebra. It helps us solve equations and learn more about relationships within mathematical operations.

Translating Verbal Phrases

Translating verbal phrases into algebraic expressions is like learning a new language. It involves converting plain English sentences into mathematical language, allowing us to work with them using algebraic rules.
Consider these steps in translation:

• Identify the operations: Words like 'sum', 'product', and 'difference' suggest specific operations (addition, multiplication, and subtraction, respectively).
• Assign symbols: Variables (like 'x') are used to represent unknown values or quantities being described.

Taking our original example, 'the sum of 4 and x' is written as '4 + x', while 'the sum of x and -8' becomes 'x - 8'. By recognizing and practicing these patterns, you can effectively translate and handle a wide range of problems.

Combining Like Terms

Combining like terms is a fundamental process in simplifying expressions. When terms in an expression share the same variables raised to the same power, they are considered 'like terms'.
Here's how to combine them:

• Identify like terms: Look for terms that have the same variable parts. In '4 + x + x - 8', the terms 'x' and 'x' are like terms.
• Combine them: Add coefficients of these like terms. So, 'x + x' becomes '2x'.

By combining like terms, you simplify the expression to a more manageable form. For example, in our expression, after combining like terms and simplifying, '4 + 2x - 8' becomes '2x - 4'. This process not only makes expressions simpler but also prepares them for further operations such as solving equations.