if x is real, solve for x, 2|x|-|x+1| = 1

We need to solve the equation \( 2|x| - |x+1| = 1 \) for real \( x \).

### Step 1: Identify critical points
The critical points occur where the expressions inside the absolute values change sign. Specifically, we need to consider the points \( x = 0 \) and \( x = -1 \). This gives us three cases to examine:
1. \( x \geq 0 \)
2. \( -1 \leq x < 0 \)
3. \( x < -1 \)

### Step 2: Solve for each case

#### Case 1: \( x \geq 0 \)
For \( x \geq 0 \), \( |x| = x \) and \( |x+1| = x + 1 \). The equation becomes:
\[
2x - (x + 1) = 1
\]
Simplifying:
\[
2x - x - 1 = 1 \quad \Rightarrow \quad x - 1 = 1 \quad \Rightarrow \quad x = 2
\]
Since \( x = 2 \) satisfies \( x \geq 0 \), it is a valid solution.

#### Case 2: \( -1 \leq x < 0 \)
For \( -1 \leq x < 0 \), \( |x| = -x \) and \( |x+1| = x + 1 \). The equation becomes:
\[
2(-x) - (x + 1) = 1
\]
Simplifying:
\[
-2x - x - 1 = 1 \quad \Rightarrow \quad -3x - 1 = 1 \quad \Rightarrow \quad -3x = 2 \quad \Rightarrow \quad x = -\frac{2}{3}
\]
Since \( x = -\frac{2}{3} \) satisfies \( -1 \leq x < 0 \), it is a valid solution.

#### Case 3: \( x < -1 \)
For \( x < -1 \), \( |x| = -x \) and \( |x+1| = -(x + 1) \). The equation becomes:
\[
2(-x) - (-(x + 1)) = 1
\]
Simplifying:
\[
-2x + x + 1 = 1 \quad \Rightarrow \quad -x + 1 = 1 \quad \Rightarrow \quad -x = 0 \quad \Rightarrow \quad x = 0
\]
However, \( x = 0 \) does not satisfy \( x < -1 \), so there is no solution in this case.

### Final Answer:
The solutions are \( \boxed{2} \) and \( \boxed{-\frac{2}{3}} \).

Would you like further details or have any questions?

Here are some related questions:
1. How do you graph absolute value equations?
2. What are other methods to solve absolute value equations?
3. Can absolute value equations have infinite solutions?
4. How can you determine the number of solutions to an absolute value equation?
5. How do you solve absolute value inequalities?

**Tip:** When solving absolute value equations, always consider the sign changes at critical points, as these determine the intervals where the equation behaves differently.

Learn how to solve the absolute value equation 2|x| - |x+1| = 1 step-by-step. Explore the critical points and cases involved in solving this equation. Suitable for advanced high school students.

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