Problem 47 - Calculus with Concepts in Calculus

Question

Solve the equation. $$ |x|=1 $$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -1\).

Step 1: Understand Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. Thus, \(|x| = 1\) implies that the number \(x\) could be either 1 unit to the right of 0 or 1 unit to the left of 0.

Step 2: Set Up Two Equations

Since \(|x| = 1\) indicates that \(x\) is 1 unit away from zero, we have two possibilities: either \(x = 1\) or \(x = -1\). These correspond to the two cases for the absolute value function where it takes the positive and negative form of the value within the bars.

Step 3: Verify the Solutions

Substitute both values into the original absolute value equation to confirm they satisfy the condition. For \(x = 1\), the calculation yields \(|1| = 1\), which is true. Similarly, for \(x = -1\), the calculation yields \(|-1| = 1\), which is also true.

Key Concepts

Solving Equations

When solving equations involving absolute values, the goal is to determine all possible values of the variable that satisfy the equation. This involves understanding how absolute values operate. In algebra, the absolute value symbol \(|x|\) signifies that you are looking at the distance of a number from 0 on a number line, without regard to direction. Thus, solving an equation like \(|x| = 1\) means finding all numbers whose distance from zero is 1.

To solve \(|x| = 1\), you first consider what numbers have an absolute value of 1. Since absolute value considers both directions on the number line, there are two potential solutions:

• One solution is when the number is 1 unit to the right of 0, which gives \(x = 1\).
• The other solution is when the number is 1 unit to the left of 0, resulting in \(x = -1\).

The process of setting up two potential solutions is crucial because absolute values can represent positive or negative scenarios that yield the same distance.

Distance from Zero

Distance from zero is a key concept in understanding absolute values. \(x\) is the number in question, and \(|x|\) demonstrates how far this number is from zero on the number line.
The equation \(|x| = 1\) can be thought of as asking, “What numbers are exactly 1 unit away from 0?”

The understanding of distance in absolute value implies ignoring the direction—whether the number is to the positive side (right) or negative side (left) of zero. Therefore, for \(|x| = 1\), \(x\) must be 1 or -1, because these are the numbers that are 1 unit away from zero on the number line.

It's important to grasp this basic interpretation as it allows you to apply the concept to more complex equations and scenarios involving distances on a number line.

Number Line

The number line is a simple but powerful tool to visualize concepts like absolute values and their solutions. A number line stretches infinitely in both directions from zero, with positive numbers to the right and negative numbers to the left.

When dealing with an equation such as \(|x| = 1\), visualizing on a number line helps you see why both \(x = 1\) and \(x = -1\) are valid solutions. By identifying these numbers on the number line, you can clearly see that both are exactly 1 unit away from zero.

• To the right, 1 is the first unit mark.
• To the left, -1 is also the first unit mark.

Understanding how to use a number line can make complex mathematical ideas more intuitive, enabling students to easily solve equations involving absolute values and other algebraic functions.