Problem 49 - Calculus with Concepts in Calculus

Question

Solve the equation. $$ |x-1|=2 $$

Step-by-Step Solution

Answer

The equation \(|x-1|=2\) has solutions \(x = 3\) and \(x = -1\).

Step 1: Understanding Absolute Value

The absolute value of a number represents its distance from zero on the number line. For any number \(|x-1| = 2\), there are two possible cases: either \(x-1 = 2\) or \(x-1 = -2\). This is because both 2 and -2 are two units away from zero.

Step 2: Solving Case 1

Consider the case where \(x - 1 = 2\). To find \(x\), add 1 to both sides: \[x - 1 + 1 = 2 + 1\] which simplifies to \(x = 3\).

Step 3: Solving Case 2

Consider the case where \(x - 1 = -2\). To find \(x\), add 1 to both sides: \[x - 1 + 1 = -2 + 1\] which simplifies to \(x = -1\).

Step 4: Combining Solutions

Both solutions \(x = 3\) and \(x = -1\) are valid for the absolute value equation \(|x-1|=2\). Thus, the complete solution is \(x = 3\) or \(x = -1\).

Key Concepts

Solving Absolute Value Equations

The goal of solving an absolute value equation, like \(|x-1| = 2\), is to find all possible values of \(x\) that make the equation true. Absolute value describes a number's distance from zero on the number line, regardless of direction. In basic terms, the expression within the absolute value signs, \(x-1\), can either be equal to the positive distance 2 or the negative distance -2 from zero. Hence, for this specific equation, we consider two separate cases:

• One where \(x-1 = 2\)
• Another where \(x-1 = -2\)

These cases represent all possible ways to get an absolute value of 2 from a number, leading us to pinpoint the precise values of \(x\) by solving each one. This approach ensures that no potential solution for \(x\) is overlooked.

Distance on the Number Line

Understanding how distance on a number line relates to absolute value is crucial in grasping absolute value equations. The absolute value of a number indicates how far the number is from zero, irrespective of direction. For \(|x-1| = 2\), this means either a shift 2 units to the right or 2 units to the left from 1.

• If we move right, we land at 3, making \(x-1 = 2\) or \(x = 3\).
• If we shift left, we find ourselves at -1, making \(x-1 = -2\) or \(x = -1\).

Both these conditions return the absolute value of 2 due to their symmetric positions around the pivot point of 1, corresponding to the nature of absolute value equations.

Solution Cases for Equations

When tackling solution cases for equations in absolute value contexts, it is important to set up equations based on each scenario the absolute expression results in. For the equation \(|x-1| = 2\), our two cases are:

• Case 1: \(x-1 = 2\)
• Case 2: \(x-1 = -2\)

Let's solve these step-by-step. First, in Case 1, we solve \(x-1 = 2\) by adding 1 to both sides, giving us \(x = 3\). Next, in Case 2, solving \(x-1 = -2\) also involves adding 1 to both sides, resulting in \(x = -1\).
Combining these solutions, \(x = 3\) and \(x = -1\) are the numbers that satisfy the initial absolute value equation. This dual approach ensures checking all mathematical possibilities delivered by the absolute value.