Okay, so I need to find the largest value of x that satisfies the equation |5x - 1| = x + 3. Hmm, absolute value equations can sometimes have two solutions, right? Because the expression inside the absolute value can be either positive or negative. But since we're asked for the largest value of x, I guess there might be two solutions and I just need to pick the bigger one. Let me think step by step.

First, I remember that when dealing with absolute value equations, you have to consider two cases: one where the expression inside the absolute value is non-negative, and another where it's negative. So in this case, the equation |5x - 1| = x + 3 can be split into two separate equations.

Case 1: 5x - 1 is non-negative. That means 5x - 1 ≥ 0. Solving this inequality, 5x ≥ 1, so x ≥ 1/5. In this case, the absolute value just becomes 5x - 1, so the equation becomes 5x - 1 = x + 3. Let me solve that.

5x - 1 = x + 3  
Subtract x from both sides: 4x - 1 = 3  
Add 1 to both sides: 4x = 4  
Divide by 4: x = 1

Okay, so x = 1 is a potential solution. But wait, we have to check if this solution satisfies the condition of Case 1, which is x ≥ 1/5. Since 1 is definitely greater than 1/5, this solution is valid.

Case 2: 5x - 1 is negative. That means 5x - 1 < 0. Solving this inequality: 5x < 1, so x < 1/5. In this case, the absolute value becomes -(5x - 1) = -5x + 1. So the equation becomes -5x + 1 = x + 3. Let me solve that.

-5x + 1 = x + 3  
Add 5x to both sides: 1 = 6x + 3  
Subtract 3 from both sides: -2 = 6x  
Divide by 6: x = -2/6 = -1/3

So x = -1/3 is another potential solution. But again, we need to check if this solution satisfies the condition of Case 2, which is x < 1/5. Since -1/3 is approximately -0.333, which is indeed less than 1/5 (0.2), this solution is also valid.

Now, we have two solutions: x = 1 and x = -1/3. The question asks for the largest value of x, so between 1 and -1/3, 1 is clearly larger. Therefore, the answer should be 1.

Wait a second, let me verify both solutions in the original equation to make sure I didn't make any mistakes.

First, testing x = 1:

Left side: |5(1) - 1| = |5 - 1| = |4| = 4  
Right side: 1 + 3 = 4  
So 4 = 4, which checks out.

Testing x = -1/3:

Left side: |5(-1/3) - 1| = |-5/3 - 1| = |-5/3 - 3/3| = |-8/3| = 8/3  
Right side: (-1/3) + 3 = (-1/3) + (9/3) = 8/3  
So 8/3 = 8/3, which also checks out.

Both solutions are valid. Therefore, the largest value of x is indeed 1.

But just to make sure I didn't miss any other possible cases or make any algebraic errors, let me go through the steps again.

Starting with the equation |5x - 1| = x + 3.

Case 1: 5x - 1 ≥ 0  
5x - 1 = x + 3  
4x = 4  
x = 1  
Check if 5(1) - 1 ≥ 0: 4 ≥ 0, which is true.

Case 2: 5x - 1 < 0  
- (5x - 1) = x + 3  
-5x + 1 = x + 3  
-6x = 2  
x = -2/6 = -1/3  
Check if 5(-1/3) - 1 < 0: -5/3 - 3/3 = -8/3 < 0, which is true.

So both cases are valid. Therefore, there are two solutions, and the larger one is 1. Hence, the answer is 1.

I think that's thorough. I considered both cases, solved each equation, checked the validity of each solution against the original conditions, and verified both solutions in the original equation. No mistakes detected. The largest value is indeed 1.
Therefore, the largest value of \( x \) that satisfies the equation \( |5x - 1| = x + 3 \) is \(\boxed{1}\).
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