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Chapter 1: Problem 128

Which integers have an absolute value equal to \(45 ?\)

Short Answer

Expert verified
The integers are 45 and -45.

Step by step solution

01

Understanding Absolute Value

Absolute value measures the distance of a number from zero on the number line, without considering direction. Therefore, both a positive and its corresponding negative number have the same absolute value. In this exercise, we want to find which integers have an absolute value of 45.
02

Set up the equation

Let the integer be represented as \( x \). The equation for this scenario is \( |x| = 45 \). Since absolute value strips the sign from the number, \( x \) can potentially be positive or negative.
03

Solving the equation for positive value

If \( x \) is positive, then \( x = 45 \). This is straightforward as the absolute value of 45 is itself.
04

Solving the equation for negative value

If \( x \) is negative, then \( x = -45 \). This is because the absolute value of \(-45\) is also 45. Net distance from zero remains the same.

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Key Concepts

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

Understanding Integer Solutions
When solving problems involving absolute values, we're often asked to find integer solutions. Integers include all whole numbers and their negative counterparts such as
  • -3, -2, -1, 0, 1, 2, 3, etc.
In this context, finding integer solutions means determining which whole numbers satisfy the given condition. For our exercise, we need whole numbers that have an absolute value of 45. An integer solution simplifies the problem within the boundaries of whole numbers, allowing you to calculate without dealing with fractions or decimals. This is why the solutions here are precisely -45 and 45.
Usage of the Number Line
The number line is a simple tool that visually represents numbers as points along a continuous line. Every integer has a position here. This line extends infinitely in both directions:
  • Positive numbers to the right of zero
  • Negative numbers to the left of zero
  • Zero sits right at the center
When we talk about the absolute value, it's about the distance from zero, not the direction. For example, both 45 and -45 are 45 units away from zero. Using the number line, you can easily see that whether you're on 45 or -45, the distance to zero is the same.
Grasping Positive and Negative Numbers
Positive and negative numbers allow us to measure quantities in opposite directions from zero. These numbers are integral to understanding mathematical contexts and real-world situations. Here are key ideas:
  • Positive numbers: greater than zero, shown to the right on a number line.
  • Negative numbers: less than zero, represented to the left.
In absolute value problems, the direction doesn鈥檛 matter; it鈥檚 the amount away from zero that counts. Therefore, knowing how positive and negative numbers work helps you understand why both -45 and 45 share the same absolute value.

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