Solve |3x+1| = 1

1. To address the equation involving the absolute value, |3x + 1| = 1, we must first recognize that the absolute value expression can yield two distinct cases. The absolute value of a number is defined as its distance from zero on the number line, which means that the expression inside the absolute value can either be equal to 1 or its negative counterpart, -1. Therefore, we can set up two separate equations to solve for the variable x: 3x + 1 = 1 and 3x + 1 = -1.

2. In the first case, we simplify the equation 3x + 1 = 1 by subtracting 1 from both sides, resulting in 3x = 0. Dividing both sides by 3 gives us the solution x = 0. In the second case, we take the equation 3x + 1 = -1 and again isolate the variable by subtracting 1 from both sides, leading to 3x = -2. Dividing by 3 in this instance yields the solution x = -2/3. Thus, we have identified two solutions to the original absolute value equation.

3. The solutions x = 0 and x = -2/3 can be verified by substituting them back into the original equation to ensure they satisfy the condition set by the absolute value. This process of solving absolute value equations is a fundamental concept in mathematics, often covered in courses such as an A Level Maths Revision Course. Mastery of these techniques is essential for students as they prepare for advanced mathematical studies and applications.