Problem 64 - Algebra 1

Question

EQUATIONS AND INEQUALITIES Match the verbal sentence with its mathematical representation. The product of 16 and \(x\) is greater than 32.

Step-by-Step Solution

Answer

The mathematical representation of the verbal sentence is \(16x>32\).

Step 1: Identify the operation

In the given sentence, 'The product of 16 and \(x\)' indicates a multiplication operation. Hence, \(16x\) will represent 'The product of 16 and \(x\)'.

Step 2: Translate 'is greater than'

'Is greater than' can be translated as the inequality symbol '>'. Thus, ’16x is greater than' can be represented as \(16x >\).

Step 3: Complete the inequality

According to the given verbal sentence, the product of 16 and \(x\) is greater than 32. Therefore, after combining all the mathematical symbols and numbers, the inequality becomes: \(16x>32\).

Key Concepts

Verbal Sentences to Equations

Verbal sentences can often seem daunting at first. However, they are simply sentences that describe mathematical operations or relationships. By breaking down these sentences, we can convert them into equations or inequalities that are easier to work with. Let's take the sentence, "The product of 16 and \(x\) is greater than 32." Here is a simple breakdown:

• Product of 16 and \(x\): The word 'product' signifies multiplication. Thus, this part of the sentence tells us to multiply 16 by \(x\), which is represented mathematically as \(16x\).
• Is greater than: This describes a comparison and is translated to an inequality symbol, which, in this case, is '>'.
• 32: The number remains the same, indicating what 16 times \(x\) is compared against.

Putting these together gives us the inequality: \(16x > 32\). By systematically breaking down the verbal sentence, it becomes much simpler to translate into an actionable mathematical statement.

Multiplication in Inequalities

When dealing with inequalities like \(16x > 32\), multiplication plays a vital role in deciphering and solving it. Understanding how multiplication applies in such scenarios can simplify the process greatly.To break it down:

• Multiplicative Relationship: The term \(16x\) shows that 16 is multiplied by an unknown value \(x\). This is the core operation defining the inequality.
• Affecting the Inequality: Any operation you do to one side of an inequality, you must do to the other. While multiplication doesn't affect the direction of the inequality, it's crucial to know this for when you solve the inequality. For example, if dividing or multiplying by a negative number, the inequality sign flips.

Understanding these rules helps maintain the inequality's integrity and leads to valid solutions when solving for \(x\). This ensures the inequality adheres to its original meaning.

Translating Sentences to Inequalities

Translating sentences to inequalities is a valuable skill that enables us to represent situations mathematically. It involves careful interpretation of words and recognizing their numeric or symbolic counterparts. In the sentence "The product of 16 and \(x\) is greater than 32," let's see how translation happens:

• Translating "The product of 16 and \(x\):" We already know that 'product' refers to multiplication, so we express this as \(16x\).
• Translating "is greater than 32:" The phrase 'is greater than' suggests an inequality where one side is larger than the other, represented by '>'. Therefore, this part becomes \( > 32\).

By combining these, we achieve the inequality \(16x > 32\). This approach helps convey a clear mathematical meaning from a worded statement. Mastery of this translation process is key to solving and understanding algebraic inequalities effectively, making it an essential skill in mathematical problem-solving.