Problem 70
Question
Express the distance between the given numbers using absolute value. Then fi nd the distance by evaluating the absolute value expression. -6 and 8.
Step-by-Step Solution
Verified Answer
The distance between -6 and 8 is 14 units.
1Step 1: Understand the Absolute Value of a Number
The absolute value of a number is the distance of that number from zero on a number line. It is denoted as \(|a|\), where \(a\) is a given number.
2Step 2: Determine the Absolute Value Expression of the Distance
The distance between two numbers (lets say \(a\) and \(b\)) on the number line is given by the absolute value of their difference, represented as \(|a - b|\) or \(|b - a|\), either one can be used as distance is always positive. As given numbers are -6 and 8, the desired absolute expression becomes \(|-6 - 8|\) or \(|8 - (-6)|\).
3Step 3: Find the Distance by Evaluating the Expression
Evaluate the either absoulte expression to find the distance. If we chose \(|-6 - 8|\), it simplifies to \(|-14|\) which returns 14 as absolute value of any number is always positive. Or if we chose \(|8 - (-6)|\), it simplifies to \(| 8 + 6 |\) resulting in \(|14|\) which is also 14. Either ways, the distance between -6 and 8 is 14.
Key Concepts
Absolute ValueNumber LineDistance Between NumbersEvaluating Expressions
Absolute Value
The concept of absolute value is fundamental to understanding how we measure distance on the number line. It represents the numerical value of a number without regard to its sign. In simpler terms, absolute value can be thought of as a number's distance from zero. For instance, both \( -3 \) and \( 3 \) have the same absolute value: \( | -3 | = | 3 | = 3 \) because both are 3 units away from zero on the number line. Whenever you see the vertical bars, like \( |a| \) where \( a \) could be any real number, it means you're looking at the absolute value of \( a \).
When evaluating the absolute value of negative numbers, it's important to remember that the result is always positive. Why? Because it reflects a distance, and distances cannot be negative. This means \( |-a| = a \) as long as \( a \) is positive. For example, \( |-7| \) gives us \( 7 \) because if you're 7 steps to the left of zero on the number line, you’re still 7 steps away regardless of direction.
When evaluating the absolute value of negative numbers, it's important to remember that the result is always positive. Why? Because it reflects a distance, and distances cannot be negative. This means \( |-a| = a \) as long as \( a \) is positive. For example, \( |-7| \) gives us \( 7 \) because if you're 7 steps to the left of zero on the number line, you’re still 7 steps away regardless of direction.
Number Line
Imagine the number line as a long horizontal road stretching infinitely in both directions, with zero at the center. Numbers to the right of zero are positive, and numbers to the left are negative. This visual helps clarify various mathematical concepts, especially when you’re asked to compare numbers or find their distance from zero or each other.
The number line not only helps us visualize the placement of numbers, but it also serves as a tool for performing addition and subtraction. Think of each movement to the right as an increase (addition) and each movement to the left as a decrease (subtraction). It's a practical way to understand how numbers interact on a basic level and is instrumental when dealing with absolute values, as we represent the distance between two points on this line.
The number line not only helps us visualize the placement of numbers, but it also serves as a tool for performing addition and subtraction. Think of each movement to the right as an increase (addition) and each movement to the left as a decrease (subtraction). It's a practical way to understand how numbers interact on a basic level and is instrumental when dealing with absolute values, as we represent the distance between two points on this line.
Distance Between Numbers
To find the distance between two numbers on the number line, we use absolute value. This distance is always a positive number and can be thought of as how many steps it would take to travel from one number to the other. The mathematical expression for distance between two numbers \( a \) and \( b \) is \( |a - b| \) or \( |b - a| \), as order doesn’t affect the outcome.
It’s crucial to comprehend that this distance is not simply the difference between the numbers, but the absolute difference. For example, the distance between \( 4 \) and \( -2 \) is \( |4 - (-2)| = |4 + 2| = |6| = 6 \) steps, and the same goes for \( |-2 - 4| = |-6| = 6 \) steps. Whether you move to the left or to the right first on the number line, you end up covering the same ground, highlighting the value of absolute measurements in math.
It’s crucial to comprehend that this distance is not simply the difference between the numbers, but the absolute difference. For example, the distance between \( 4 \) and \( -2 \) is \( |4 - (-2)| = |4 + 2| = |6| = 6 \) steps, and the same goes for \( |-2 - 4| = |-6| = 6 \) steps. Whether you move to the left or to the right first on the number line, you end up covering the same ground, highlighting the value of absolute measurements in math.
Evaluating Expressions
When it comes to algebra, evaluating expressions is a way of finding out what they are worth by performing the operations they contain. To do this, you substitute the values given for any variables and calculate the result, following the rules of arithmetic and considering the order of operations: parentheses, exponents, multiplication and division, and addition and subtraction (PEMDAS).
Evaluation becomes particularly interesting with absolute value expressions. Since the absolute value itself is an operation, it must be performed after any calculations inside the absolute value bars, but before any outside. For the exercise at hand evaluating \( |-6 - 8| \) involves recognizing it as \( |-14| \) and then interpreting the absolute value, which gives us \( 14 \) because the distance from \( -14 \) to \( 0 \) on the number line is exactly \( 14 \) units. The process of evaluating absolute value expressions encourages students to think about the mathematical implications of distance and magnitude, reinforcing their understanding of absolute values and their role in analyzing expressions.
Evaluation becomes particularly interesting with absolute value expressions. Since the absolute value itself is an operation, it must be performed after any calculations inside the absolute value bars, but before any outside. For the exercise at hand evaluating \( |-6 - 8| \) involves recognizing it as \( |-14| \) and then interpreting the absolute value, which gives us \( 14 \) because the distance from \( -14 \) to \( 0 \) on the number line is exactly \( 14 \) units. The process of evaluating absolute value expressions encourages students to think about the mathematical implications of distance and magnitude, reinforcing their understanding of absolute values and their role in analyzing expressions.
Other exercises in this chapter
Problem 70
In Exercises 67–82, find each product. $$(3 x-y)(2 x+5 y)$$
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Simplify the radical expressions in Exercises \(67-74\) if possible. $$\sqrt[3]{x^{5}}$$
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Factor completely, or state that the polynomial is prime. $$x^{3}+2 x^{2}-9 x-18$$
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Write each number in decimal notation without the use of exponents. $$7.9 \times 10^{-1}$$
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