Problem 130 - Elementary Algebra

Question

Determine the unknown. $$ |?|=1 $$

Step-by-Step Solution

Answer

The unknown can be 1 or -1.

Step 1: Understand the Absolute Value

Absolute value represents the distance of a number from 0 on the number line, regardless of direction. If an absolute value is given as 1, then the number can either be 1 unit to the right of 0 or 1 unit to the left of 0.

Step 2: Set up Possible Equations

Given the equation \(|?| = 1\), interpret it as two possible simple equations: either the number is 1 or the number is -1, due to the definition of absolute value. Thus, we have two possibilities: \(? = 1\) or \(? = -1\).

Step 3: Solve the Equations

For \(? = 1\), the solution is straightforwardly 1. For \(? = -1\), the solution is straightforwardly -1. So, the absolute value equation \(|?| = 1\) yields two potential solutions: 1 and -1.

Key Concepts

Understanding the Number Line

The number line is a simple yet powerful tool for understanding mathematical concepts. It is essentially a straight line that represents numbers in a sequential order. Each point on this line corresponds to a real number. Zero, often found at the center, acts as a reference point.

• Positive numbers are located to the right of zero.
• Negative numbers are located to the left.

Understanding the number line helps us visualize the concept of distance, which is crucial for grasping absolute values. When solving absolute value problems, like \(|?| = 1\), we see that a number is just a certain distance away from zero. In this case, that distance is 1.

Solving Equations Involving Absolute Values

Equations with absolute values can sometimes be tricky, but they follow a simple logic. When you encounter an absolute value equation like \(|?| = 1\), it means we are looking for numbers whose distance from zero is 1. Hence, this equation can be split into simpler equations.

• The first possibility is that the unknown is 1, since \(|1| = 1\).
• The second is that the unknown is -1, since \(|-1| = 1\).

Thus, there are typically two solutions, as each reflects one of the possible positions on the number line. By breaking it down, we see that these absolute value equations always yield outcomes reflecting that distance.

Visualizing Distance from Zero

Distance from zero is a key idea when it comes to understanding absolute values. This distance is always a positive number, as it measures how far a number is from zero on the number line.

• For example, the distance of both 1 and -1 from zero is 1 unit.

The concept of distance being always positive is why absolute value equations often have two solutions. Negative or positive numbers at the same absolute distance from zero yield the same absolute value. This helps in visualizing and solving problems where you'll need to find all numbers at a defined distance from the center of the number line.