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Chapter 2: Problem 53

Provide a number line sketch with your answer. Find two integers on the number line that are 2 units away from the integer 2 .

Short Answer

Expert verified
The integers are 0 and 4.

Step by step solution

01

Understand the Problem

We need to find two integers that are each 2 units away from the integer 2. This means we have to find two integers, say \( x \), such that the distance between \( x \) and 2 is 2 units.
02

Set Up the Equation

The mathematical expression for the distance being 2 is given by \(|x - 2| = 2 \). This equation represents the distance from \( x \) to 2 being equal to 2.
03

Solve the Equation

The equation \(|x - 2| = 2 \) can be solved by splitting it into two separate cases: 1. \( x - 2 = 2 \) which simplifies to \( x = 4 \). 2. \( x - 2 = -2 \) which simplifies to \( x = 0 \). Thus, the two integers are 4 and 0.
04

Draw the Number Line

Draw a number line, and mark 0, 2, and 4 on it. - Find integer 2. It is at the center of the segment. - Move 2 units to the right of 2, landing on 4. - Move 2 units to the left of 2, landing on 0. - The integers 0 and 4 are both 2 units away from 2.

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Key Concepts

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

Integer
An integer is a whole number that can be positive, negative, or zero. Integers do not have fractions or decimals and are one of the basic building blocks of mathematics. For example:
  • Numbers like -3, 0, 7 are integers.
  • Expressions like \( \frac{1}{2} \), or 3.5 are not integers because they are either fractions or decimals.
Integers are important when working with number lines, which often depict whole numbers in order for easy visualization. On a number line, integers are spaced equally apart, and this helps us easily understand concepts like counting and measuring distance between points. Understanding integers is crucial when solving problems involving number lines, as they allow us to locate and identify specific points.
Distance
Distance on a number line refers to how far apart numbers are from each other. It is always expressed as a positive number because we are measuring how much space there is between two numbers, not a direction. For instance:
  • If we are to find the distance between the integers 2 and 4 on the number line, we are essentially counting the number of steps needed to get from 2 to 4.
  • This makes the distance between them as being simply 2 units.
Even when dealing with numbers that are negative, distance on a number line remains positive. Thus, talking about the distance between 2 and -3 would mean counting each step you take from 2, passing by 1, 0, -1, -2, to finally reach -3, resulting in a distance of 5 units. This notion underlines the fundamental concept of distance: looking at the number of units apart two numbers are, regardless of their position on the number line.
Absolute Value
Absolute value is a concept referring to the non-negative value of a number without regard to its sign. When dealing with number lines, absolute value helps in understanding how far a number is from zero. Here's how it works:
  • The absolute value of a positive number, like 5, is 5 because it is already non-negative.
  • The absolute value of a negative number, like -5, is also 5. Essentially, the absolute value function drops the negative sign and looks at the number's magnitude, or how "big" the number is compared to zero.
In mathematical notation, the absolute value of \( x \) is denoted as \( |x| \). When we return to our number line example about finding numbers 2 units away from 2, the absolute value notation \( |x - 2| = 2 \) is saying that we're measuring a distance of 2 units from 2, without regard to whether we are moving left or right. Hence the solutions found, 0 and 4, are those points which lie exactly at that specified distance from 2.

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