Problem 69 - College Algebra

Question

Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. \(-2\) and 5

Step-by-Step Solution

Answer

The distance between -2 and 5 is 7.

Step 1: Expression setup

Set up the absolute value expression to represent the distance between the numbers. To do this, subtract one number from the other and take the absolute value of the result. The expression is \(|(-2) - 5|\).

Step 2: Calculate inside the absolute value

Calculate the difference inside the absolute value symbol. In this case, \((-2) - 5 = -7\). So, the expression becomes \(|-7|\).

Step 3: Evaluate Absolute Value

Finally, evaluate the absolute value of -7. Absolute value makes any negative number into a positive, so \(|-7|=7 \).

Key Concepts

Evaluating Absolute Value

Understanding the absolute value of a number is a foundational concept in algebra that refers to the distance of that number from zero on a number line. In its simplest form, the absolute value function, denoted by the symbols | |, turns any negative number into its positive counterpart. For example, if you see the expression \( |{-7}| \), the absolute value of -7 is 7 because it is 7 units away from zero.

When evaluating the absolute value of an algebraic expression, such as \( |{-2} - 5| \), you would first perform the subtraction inside the absolute value symbols to find \( (-2) - 5 = -7 \). Then, apply the absolute value to get the result, \( |-7| = 7 \). It is important to remember that no matter the initial sign of the number inside the absolute value, the outcome is always non-negative. This function is fundamental in various applications, including determining the distance between two points and solving equations involving absolute value.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations (such as addition, subtraction, multiplication, and division). They are essentially a shorthand way to represent a computation or to encode a pattern. For instance, \( |{-2} - 5| \) is an algebraic expression that involves the operation of subtraction as well as the function of absolute value.

An essential skill in algebra is learning how to simplify and evaluate these expressions. Simplifying might involve combining like terms or using the distributive property, while evaluating would usually mean substituting numbers for variables and then completing the operations in the proper order, typically guided by the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

One powerful aspect of algebraic expressions is their use in modeling real-world situations. They can describe relationships and changes between quantities, making them a versatile tool in various fields of study.

Distance Between Two Points

In mathematics, distance between two points refers to the numerical measurement of how far apart these points are on a number line or in a geometric space. When dealing with a number line, finding the distance between two points is as simple as evaluating the absolute value of the difference between the coordinates of these points. The absolute value is used because distance is always a positive quantity, or non-negative to be precise.

For example, to find the distance between the points -2 and 5 on a number line, you subtract one point from another to get \( (-2) - 5 \) and then take the absolute value to remove any negative sign, resulting in a distance of 7 units. This technique extends into multiple dimensions as well—for higher-dimensional spaces, the distance formula derived from Pythagoras' Theorem is commonly used, which involves square roots and squares of differences between coordinates of the points. Nonetheless, the fundamental idea remains: distance is always measured as a non-negative difference.