Problem 63 - Elementary Algebra Within Reach

Question

In Exercises 63-68, translate the verbal phrase into an algebraic expression. Simplify the expression. $$ x \text { times the sum of } x \text { and } 3 $$

Step-by-Step Solution

Answer

The simplified form of 'x times the sum of x and 3' is \(x^2 + 3x\).

Step 1: Translate the verbal phrase into an algebraic expression

The verbal phrase 'x times the sum of x and 3' can be translated into the algebraic expression \(x \cdot (x + 3)\), where 'times' represents multiplication and 'the sum of x and 3' is represented by \(x + 3\) within brackets.

Step 2: Apply the distributive property to simplify the expression

The distributive property states that \(a(b + c) = ab + ac\). Therefore, we can distribute \(x\) across \(x + 3\) to simplify the expression. This results in the new expression \(x \cdot x + x \cdot 3\).

Step 3: Perform the multiplication operations

Now, perform the multiplication operation in the expression found in step 2. The result will be \(x^2 + 3x\).

Key Concepts

Verbal Phrase Translation

Verbal phrase translation is a foundational skill in algebra that involves converting everyday language into mathematical expressions. When presented with phrases like "x times the sum of x and 3," it is important to recognize key words that indicate specific operations. For example:

• "Times" indicates multiplication.
• "Sum" suggests addition.

In the phrase "x times the sum of x and 3," "times" implies that x is multiplying an entire quantity, specifically "the sum of x and 3." The words "sum of x and 3" tell us that we need to first add x and 3 together. Therefore, we enclose this operation in parentheses: \(x + 3\). The entire verbal expression translates to \(x \cdot (x + 3)\), using a multiplication symbol to indicate that x multiplies the sum.

Distributive Property

The distributive property is a vital rule in algebra used to simplify expressions. It states that multiplication distributed over addition (or subtraction) can be expressed as \(a(b + c) = ab + ac\). Applying this rule enables us to simplify expressions that involve parentheses.
When we translate the verbal phrase to \(x \cdot (x + 3)\), we utilize the distributive property to simplify it:

• Step 1: Multiply the first term within the parentheses by x. This gives us \(x \cdot x = x^2\).
• Step 2: Multiply the second term within the parentheses by x. This yields \(x \cdot 3 = 3x\).

Put these results together, and we obtain the simplified expression \(x^2 + 3x\). This approach not only simplifies expressions but also sets a foundation for solving more complex algebraic equations.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process makes it easier to understand and work with the expressions. After applying the distributive property, the expression \(x \cdot (x + 3)\) is rewritten as \(x^2 + 3x\). This is what we call 'simplifying' because:

• We have combined like terms and completed necessary operations.
• The expression is concise and displays the mathematical relationships clearly.

This simpler form, \(x^2 + 3x\), is more manageable for further algebraic operations such as solving equations, graphing, or substituting values. As you practice simplifying expressions, focus on methodically applying mathematical operations and combining terms where applicable, to make your work cleaner and easier to follow.