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Chapter 2: Problem 6

Explain why \(13-x\) is not a correct translation of "13 less than a number:"

Short Answer

Expert verified
The correct translation of '13 less than a number' is \(x - 13\), not \(13 - x\).

Step by step solution

01

Understanding the Phrase

Interpret the phrase '13 less than a number.' This phrase means that 13 is being subtracted from some unknown number (let's denote this unknown number as x).
02

Setting Up the Equation

Set up the appropriate mathematical representation. If 13 is less than a number (x), it means subtracting 13 from x. Thus, the correct expression is: \[ x - 13 \]
03

Why the Given Expression is Incorrect

Analyze the given expression \(13 - x\). This suggests that x is being subtracted from 13, which does not align with the phrase '13 less than a number.'
04

Conclusion

In conclusion, the correct translation of '13 less than a number' is \(x - 13\), not \(13 - x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Phrase Interpretation
Understanding the language used in algebra problems is crucial. When you read a phrase like '13 less than a number,' you need to break it down:
  • 'A number' refers to an unknown variable, which we often call \( x \).

  • 'Less than' indicates subtraction.

Therefore, '13 less than a number' means that we take the number (\( x \)) and subtract 13 from it. Always look for key terms like 'more than,' 'less than,' 'sum,' and 'product' to figure out what operations to use.
Setting Up Equations
Translating a phrase into an equation can be simplified by following a series of steps:
  • Identify the unknown quantity and let it be a variable, usually \( x \).

  • Translate operations from words to mathematical symbols.

In our example, '13 less than a number' means we start with a number (\( x \)) and subtract 13, so we set up the equation as \[ x - 13 \]. Always ensure the order and operations match the original phrase.
Common Mistakes in Algebra
Many students struggle with translating phrases correctly due to small but significant errors. Common mistakes include:
  • Reversing the order of terms, like misinterpreting '13 less than \( x \)' as \( 13 - x \).

  • Confusing 'less than' with 'minus,' not considering that 'less than' often means the number comes after the subtraction.

To avoid these errors, focus on understanding the meaning behind the words and always double-check your interpretation.
Subtraction in Algebra
Subtraction is a fundamental operation in algebra, but it can be tricky when interpreting phrases:
  • When you see 'less than,' it usually means you need to subtract the later term from the initial term.

  • For instance, '13 less than \( x \)' translates to \( x - 13 \).

Unlike simple numbers, in algebra, the order of subtraction matters a lot. Always ensure that you closely follow the phrase's meaning to set up the correct expression.

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