Problem 14 - Elementary Algebra

Question

Determine the unknown. \(|?|=0\)

Step-by-Step Solution

Answer

The unknown is 0.

Step 1: Understanding Absolute Value

Absolute value represents the distance a number is from zero on a number line. It is always non-negative.

Step 2: Interpreting the Absolute Value Equation

Given \(|?|=0\), this means we are looking for a number whose absolute value is 0. This implies that the number itself must be 0 because the only number that is 0 units away from 0 on the number line is 0 itself.

Step 3: Solving the Equation

Since the absolute value of 0 is 0, set the unknown equal to 0. Therefore, the solution to the equation \(|?|=0\) is \(x = 0\).

Key Concepts

Number Line

A number line is a visual representation of numbers placed at equal intervals along a straight line. It's a powerful tool that helps us understand numerical concepts we encounter in math.

• Numbers to the right of zero are positive.
• Numbers to the left of zero are negative.
• Zero is the central point, which neither has a positive nor negative value.

Understanding a number line is crucial when dealing with absolute value because it helps in visualizing and determining the position of numbers relative to zero. By using a number line, we can easily see how far away a particular number is from zero, which is the essence of absolute value.

Distance from Zero

In mathematics, the concept of distance from zero is expressed through absolute value. Absolute value, depicted by vertical bars, measures how far a number is from zero on the number line.
For example, consider

• the absolute value of both 3 and -3 is 3
• because both numbers are 3 units away from 0.

Absolute values are always non-negative because distance itself cannot be negative. In the given equation \(|?| = 0\), we are asked to find which number is 0 units away from zero. On the number line, only the number zero is at a distance of zero from itself.

Solving Equations

Solving absolute value equations involves finding a number that fits the condition set by the absolute value. For the equation \(|?|=0\), we need to determine which number has an absolute value of zero.

Steps to Solve

• Identify the Equation: In this case, the equation is \(|x|=0\).
• Interpret the Meaning: Here, we need to find a number whose absolute value equals zero.
• Determine the Solution: Since only zero has an absolute value of zero, we conclude \(x = 0\).

Solving such simple absolute value equations provides the groundwork for tackling more complex problems, reinforcing the understanding that absolute value is always about the distance from zero on a number line.