Final answer:
The absolute value equation |3x + 1| = 1 has two solutions, x = 0 and x = -2/3, obtained by considering the cases when the expression inside the absolute value is non-negative and when it is negative.
Step-by-step explanation:
To solve the absolute value equation |3x + 1| = 1, we need to consider the two possible cases for the absolute value expression.
- Case 1: If the expression inside the absolute value is non-negative (3x + 1 ≥ 0), then |3x + 1| = 3x + 1. Setting this equal to 1 gives us the equation 3x + 1 = 1. Subtracting 1 from both sides yields 3x = 0, leading to x = 0.
- Case 2: If the expression inside the absolute value is negative (3x + 1 < 0), then |3x + 1| = -(3x + 1). Setting this equal to 1 gives us the equation -(3x + 1) = 1. Multiplying both sides by -1 yields 3x + 1 = -1. Subtracting 1 from both sides gives us 3x = -2, resulting in x = -2/3.
Therefore, the two solutions to the equation |3x + 1| = 1 are x = 0 and x = -2/3.