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Chapter 2: Problem 25

Translate each phrase; then simplify. Subtract 17 from \(-25 .\)

Short Answer

Expert verified
-42

Step by step solution

01

Understand the Phrase

The phrase 'Subtract 17 from -25' requires translating the words into a mathematical expression. In this case, 'subtract' indicates a subtraction operation, and 'from' implies the number after 'from' is the starting point. Thus, the expression would be: i.e., -25 - 17.
02

Simplify the Expression

Now that we have translated the phrase into a mathematical expression, the next step is to simplify it. The expression is \(-25 - 17\).When you subtract 17 from -25, you are essentially doing: \[-25 - 17 = -25 + (-17)\].Negative numbers are being added, which is equivalent to moving further left on the number line. Thus, \(-25 + (-17) = -42\).

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Key Concepts

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

Negative Numbers: Understanding the Basics
Negative numbers are numbers that are less than zero. They are usually represented with a minus sign (e.g., \(-1, -2, -3,\ldots\)). Understanding negative numbers is crucial for performing integer operations like subtraction.
  • Negative numbers denote values below zero, often used in contexts such as temperature or debts.
  • When comparing, negative numbers are always less than positive numbers.
  • Moving left on the number line represents adding negative numbers, while going right represents subtracting them.
When working with negative numbers, remember that subtracting a negative number is like adding its positive counterpart. For example, subtracting \(-3\) is equal to adding \(+3\). Understanding this can help you simplify many arithmetic operations involving negative integers.
Subtraction with Negative Numbers
Subtraction itself is a fundamental arithmetic operation but becomes more intriguing when performed with negative numbers.
  • The phrase 'subtract \(b\) from \(a\)' means you take \(b\) away from \(a\), represented as \(a - b\).
  • To subtract a positive number from a negative one, it often results in a more negative number, since you're moving further away from zero on the number line.
  • Mathematically, \(-25 - 17\) can be rewritten as \(-25 + (-17)\), which is equivalent to moving further into the negatives.
This concept demonstrates the importance of the order of operations and clarifies why understanding how subtraction and addition interact is essential when dealing with negative numbers.
The Number Line: Visualizing Integer Operations
A number line is a straightforward yet powerful tool for visualizing arithmetic operations, especially involving negatives. It consists of a horizontal line marked with numbers at equal intervals, extending infinitely in both positive and negative directions.
  • Each point on the number line represents an integer. Moving to the right indicates increasing numbers, while moving left shows decreasing values.
  • Negative numbers are shown to the left of zero, reflecting their value being less than zero.
  • When you perform \(-25 - 17\), visually, you start from the point marked \(-25\) and move 17 units to the left, landing at \(-42\).
By using the number line, you gain a clear visual understanding of how integers interact, whether through addition, subtraction, or other operations. It serves as an excellent tool for solving and checking computations involving integers.

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