Problem 72 - Algebra and Trigonometry

Question

Express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. $$-26\quad and\quad -3$$

Step-by-Step Solution

Answer

The distance between -26 and -3 is 23.

Step 1: Write Absolute Value Expression

To find the distance between two numbers on a number line, subtract the smaller number from the larger one. Then make use of the absolute value to negate any potential negative result. Hence, the absolute value expression between -26 and -3 is expressed as \( |-26 - (-3)|\).

Step 2: Simplify the Expression

Simplify inside the absolute value bracket first, before proceeding. This results in \(|-26 + 3|\). Adding those values gives us \(-23\). Therefore, expression simplifies to \(|-23|\).

Step 3: Evaluate the Absolute Value

Now evaluate the absolute value of \(-23\). Absolute value is a numerical value of a quantity without regard to its sign. So, \(|-23|\) simply becomes \(23\).

Key Concepts

Evaluating Absolute Value

Understanding how to evaluate the absolute value of a number is essential for solving many types of mathematical problems. The absolute value of a number refers to its distance from zero on a number line, regardless of direction. Think of it as the 'numerical value' without any concern about whether it's positive or negative. For example, the absolute value of both \( -4 \) and \( 4 \) is \( 4 \), because both points are four units away from zero. To find the absolute value of a number, you simply remove any negative sign in front of the number, making it positive. This makes absolute value a very helpful tool in representing distances that cannot be negative by nature.

Number Line Distance

A number line is a visual representation of numbers laid out in a straight line where each point corresponds to a number. It is immensely useful to think of distances between numbers in terms of their positions on the number line. The concept of number line distance is akin to measuring a straight path between two points: regardless of which direction you travel, the distance remains the same. When you are calculating the distance between two points, therefore, the direction (whether one number is to the left or right of the other) is irrelevant. All that matters is how many units separate them. This principle is perfectly encapsulated in the concept of absolute value.

Absolute Value Expression

An absolute value expression wraps a number or an algebraic expression within vertical bars, such as \( |x| \) or \( |-26 - (-3)| \). This notation signifies that we're interested in finding the non-negative value of what's inside, no matter whether the initial value is positive or negative. Absolute value expressions come in handy when dealing with distances, because they translate the idea of ignoring direction into simple arithmetic. When you work with these expressions, you'll often need to simplify what's inside the bars before proceeding to evaluate the absolute value, similar to how you would simplify any expressions in parentheses first.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves performing all possible operations within an expression to arrive at the simplest form. When dealing with absolute value expressions, the process begins inside the absolute value bars. You should combine like terms, resolve any parentheses, and perform addition or subtraction as needed. Once the expression within the absolute value is as simple as it can be, you can evaluate the absolute value. This means taking the simplified number and, if it's negative, converting it to positive, because, remember, absolute value is always non-negative. Through simplification, complex-looking problems are often reduced to very basic arithmetic operations.