Solve each equation. |3x - 1 | = 2

Verified step by step guidance

1. Identify that the equation involves an absolute value: \(|3x - 1| = 2\).
2. Understand that the absolute value equation \(|A| = B\) implies two possible equations: \(A = B\) and \(A = -B\).
3. Set up the first equation without the absolute value: \(3x - 1 = 2\).
4. Set up the second equation by negating the right side: \(3x - 1 = -2\).
5. Solve each equation separately for \(x\) to find the possible solutions.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number 'a', the absolute value is denoted as |a| and is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. Understanding absolute value is crucial for solving equations that involve it, as it leads to two possible cases based on the definition.

Linear Equations

A linear equation is an equation of the first degree, meaning it involves variables raised only to the power of one. In the context of the given equation, solving for 'x' involves isolating the variable on one side of the equation. Recognizing that the absolute value equation can be split into two separate linear equations is essential for finding all possible solutions.

Case Analysis

Case analysis is a method used to solve problems that have multiple scenarios or conditions. In the context of absolute value equations, it involves setting up separate equations for each case derived from the absolute value definition. For the equation |3x - 1| = 2, this means solving both 3x - 1 = 2 and 3x - 1 = -2 to find all potential solutions for 'x'.