Problem 12 - Fundamentals of Mathematics

Question

Use the algebraic definition of absolute value to find the following values. $$ -|1| $$

Step-by-Step Solution

Answer

The value is -1.

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means it is always a non-negative number. The absolute value of a number \( a \) is denoted as \( |a| \). For example, \( |1| = 1 \) and \( |-1| = 1 \).

Step 2: Calculate the Absolute Value

Using the definition of absolute value, calculate \(|1|\). Since 1 is already positive, \(|1| = 1\).

Step 3: Apply the Negative Sign

The problem asks for \(-|1|\), which involves applying a negative sign to the absolute value we found in Step 2. Thus, \(-|1| = -1\).

Key Concepts

Algebraic Definition

The algebraic definition of absolute value is essential to grasping how it operates in algebra. The absolute value of a number is defined as the non-negative value of a number without regard to its sign. To put it more simply, if you have a number \(a\), the absolute value \(|a|\) is always that number if it's positive, or its positive equivalent if it's negative. This means:

• \(|a| = a\) if \(a \geq 0\)
• \(|a| = -a\) if \(a < 0\)

This simple definition allows us to handle numbers and their distances from zero effectively in various problems. It's like measuring the length of a string—it's always positive, no matter how you look at it.

Number Line

Visualizing the absolute value on a number line can greatly simplify understanding. A number line is like a ruler that goes infinitely in both directions with zero at the center. Positive numbers are to the right of zero and negative numbers are to the left. When you think about absolute value on this line, you're essentially ignoring the direction and just measuring distance from zero. For example, both \( 3 \) and \( -3 \) are three units away from zero. So, their absolute values are the same. Seeing numbers in this spatial context helps many students internalize why absolute values strip numbers of their direction, leaving only their size.

Negative Numbers

Negative numbers can be tricky, especially when considering absolute values. Usually, negative numbers lie to the left of zero on a number line and can signify debts, temperatures below zero, or any measure less than zero in context. In the context of absolute values, the negative sign is ignored.Applying the absolute value function to a negative number makes it positive, because we're interested in how far from zero the number is, not its direction. For example, \(|-5| = 5\). This can sometimes trip people up, but just remember that absolute value only cares about distance.

Problem Solving

Problem solving with absolute values requires careful attention to both the computations and the expressions of the problems. One common situation is finding the absolute value first and then applying operations like negation afterward, as seen with expressions like \(-|a|\).Consider the exercise provided:

• First, find \(|1|\). Since 1 is positive, \(|1| = 1\).
• Then, apply the negative sign: \(-|1| = -1\).

By breaking the problem into smaller manageable steps and referring back to the algebraic definitions, students can confidently solve more complex expressions. Remember, when problem-solving, practice like this helps reinforce understanding and skill.