Solve the Absolute Value Equation: ||x-1| + x + 1| = 4

To solve the equation $||x-1| + x + 1| = 4$, we need to consider multiple cases based on the conditions affecting the absolute value expressions. For such cases, the key points to assess are where the internal expressions become zero or change signs. Let's start solving the problem step-by-step:

Step 1: Consider the critical point $x = 1$ where the expression $|x-1|$ changes from zero or negative to positive. We handle separate cases surrounding this point.

• Case 1: $x \leq 1$

• Case 2: $x > 1$

Case 1: $x \leq 1$
In this interval, $x - 1 \leq 0$, so $|x-1| = -(x-1) = -x + 1$.

Now we consider the absolute value $||x-1| + x + 1|$:

• $||-x + 1 + x + 1|$ simplifies to $|2| = 2$, which is not equal to 4. Thus, no solutions in this interval.

Case 2: $x > 1$
In this interval, $x - 1 > 0$, so $|x-1| = x - 1$.

Now the expression becomes:

• $||(x - 1) + x + 1| = |2x| = 2x|$. Setting this equal to 4 gives $2x = 4$ or $x = 2$.

Since $x = 2$ falls within the interval $x > 1$, it satisfies the conditions for this case.

We've determined that the solution to the equation is $x = 2$, which satisfies $||x-1| + x + 1| = 4$.

Therefore, after checking the conditions for both cases and ensuring that the solution adheres to the interval restrictions, the correct solution is $x = 2$.

The correct answer is choice 2: $x = 2$.