We are asked to solve the equation:

$|x| + |x - 1| + |x + 1| = 2$

Step-by-step solution:

To solve this, we must analyze different cases based on the values of $x$ because the absolute value function behaves differently depending on whether its argument is positive or negative.

Case 1: $x \geq 1$

In this case, all expressions inside the absolute values are non-negative:

• $|x| = x$
• $|x - 1| = x - 1$
• $|x + 1| = x + 1$

Substituting these into the equation:

$x + (x - 1) + (x + 1) = 2$

Simplifying:

$x + x - 1 + x + 1 = 2$ $3x = 2 \quad \Rightarrow \quad x = \frac{2}{3}$

However, $x = \frac{2}{3}$ does not satisfy the condition $x \geq 1$. Therefore, there is no solution in this case.

Case 2: $-1 \leq x < 1$

In this interval:

• $|x| = x$
• $|x - 1| = 1 - x$
• $|x + 1| = x + 1$

Substituting into the equation:

$x + (1 - x) + (x + 1) = 2$

Simplifying:

$x + 1 - x + x + 1 = 2$ $x + 2 = 2 \quad \Rightarrow \quad x = 0$

Thus, $x = 0$ is a solution.

Case 3: $x < -1$

In this case, all expressions inside the absolute values are negative:

• $|x| = -x$
• $|x - 1| = -x + 1$
• $|x + 1| = -x - 1$

Substituting into the equation:

$-x + (-x + 1) + (-x - 1) = 2$

Simplifying:

$-x - x + 1 - x - 1 = 2$ $-3x = 2 \quad \Rightarrow \quad x = -\frac{2}{3}$

Since $x = -\frac{2}{3}$ does not satisfy the condition $x < -1$, there is no solution in this case.

Final Solution:

The only solution is:

$x = 0$

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Would you like a more detailed explanation of any step? Here are some follow-up questions:

1. How does the absolute value function behave in different cases?
2. Why was it necessary to split the problem into three cases?
3. Can you show another example of solving absolute value equations?
4. How can we check if a solution satisfies the original equation?
5. What happens if we have more than three intervals for a similar problem?

Tip: When solving absolute value equations, always check each solution in the context of the case it was derived from.