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

Verified step by step guidance

1. Recognize that the equation involves an absolute value: \(|3x - 1| = 2\). Recall that \(|A| = B\) means \(A = B\) or \(A = -B\) when \(B \geq 0\).
2. Set up two separate equations based on the definition of absolute value: \$3x - 1 = 2\( and \)3x - 1 = -2$.
3. Solve the first equation \$3x - 1 = 2\( by adding 1 to both sides: \)3x = 3\(, then divide both sides by 3 to isolate \)x$.
4. Solve the second equation \$3x - 1 = -2\( by adding 1 to both sides: \)3x = -1\(, then divide both sides by 3 to isolate \)x$.
5. Write the solution set as the values of \(x\) found from both equations.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For an expression |A| = B, where B ≥ 0, the equation splits into two cases: A = B or A = -B.

Solving Linear Equations

Solving linear equations involves isolating the variable on one side using inverse operations like addition, subtraction, multiplication, or division. This process helps find the value(s) of the variable that satisfy the equation.

Checking Solutions in Absolute Value Equations

After solving absolute value equations, it is important to check each solution by substituting back into the original equation. This ensures that extraneous solutions, which may arise from the split cases, are identified and discarded if necessary.