Final answer:
To solve the equation |3x-4| = |x|, we consider cases where the absolute value expressions equal each other or their opposites, resulting in two possible solutions, x = 1 and x = 2. Both cases need to be solved, and all true solutions checked against the original equation.
Step-by-step explanation:
To solve the equation |3x-4| = |x|, we need to consider the different cases where each absolute value expression will take on positive and negative values, and then solve the resulting linear equations. An absolute value equation can lead to two possible equations because the expression inside the absolute value can be equal to the quantity on the other side or its opposite.
Case 1: Both expressions are positive or both are negative, which means 3x-4 = x or 3x-4 = -x. Solving these, we get 2x = 4 or 4x = 4, which simplifies to x = 2 or x = 1, respectively.
Case 2: One expression is positive, and the other is negative, meaning 3x-4 = -x or -(3x-4) = x. Solving these, we get x = 4/4 = 1 or x = 4/2 = 2, respectively.
However, the second case does not provide new solutions as they repeat the solutions from the first case. By checking these answers, we find that both are true solutions to the equation. Hence, the solution to the absolute value equation is x = 1 and x = 2.