6. Let f(x) = x + |2x − 1|. Find all solutions to the equation.

To solve the equation involving the function \( f(x) = x + |2x - 1| \), we first need to understand how the absolute value affects the expression. The absolute value function \( |2x - 1| \) changes based on whether \( 2x - 1 \) is positive or negative.

### Step-by-step Solution:

1. **Identify the critical point for the absolute value expression**:

   The expression inside the absolute value is \( 2x - 1 \). The absolute value changes behavior at the point where \( 2x - 1 = 0 \).

   \[
   2x - 1 = 0 \implies x = \frac{1}{2}
   \]

   So, we need to consider two cases: \( x \geq \frac{1}{2} \) and \( x < \frac{1}{2} \).

2. **Case 1: \( x \geq \frac{1}{2} \)**

   When \( x \geq \frac{1}{2} \), \( 2x - 1 \geq 0 \), so the absolute value \( |2x - 1| = 2x - 1 \).

   The function \( f(x) \) becomes:
   \[
   f(x) = x + (2x - 1) = 3x - 1
   \]

   The equation to solve is \( 3x - 1 = 0 \).

   Solving for \( x \):
   \[
   3x - 1 = 0 \implies 3x = 1 \implies x = \frac{1}{3}
   \]

   Since \( x = \frac{1}{3} \) does not satisfy the condition \( x \geq \frac{1}{2} \), this is **not a valid solution** for this case.

3. **Case 2: \( x < \frac{1}{2} \)**

   When \( x < \frac{1}{2} \), \( 2x - 1 < 0 \), so the absolute value \( |2x - 1| = -(2x - 1) = -2x + 1 \).

   The function \( f(x) \) becomes:
   \[
   f(x) = x + (-2x + 1) = -x + 1
   \]

   The equation to solve is \( -x + 1 = 0 \).

   Solving for \( x \):
   \[
   -x + 1 = 0 \implies x = 1
   \]

   Since \( x = 1 \) does not satisfy the condition \( x < \frac{1}{2} \), this is **not a valid solution** for this case either.

### Conclusion:

There are **no valid solutions** to the equation \( f(x) = 0 \) for \( f(x) = x + |2x - 1| \).

Would you like more details or have any questions about this?

Here are 5 related questions to consider:

1. How would the solution change if the function was \( f(x) = x - |2x - 1| \)?
2. What are the properties of absolute value functions that affect their behavior in equations?
3. How do we solve equations involving multiple absolute value expressions?
4. What happens to the solutions if the function is \( f(x) = x + |2x - 3| \)?
5. Can you explain why we have to consider different cases when solving absolute value equations?

**Tip:** When solving equations with absolute values, always consider the points where the expression inside the absolute value changes sign.

Learn how to solve the equation f(x) = x + |2x - 1| step-by-step with detailed explanations. This problem involves understanding absolute value functions and solving equations. Suitable for advanced high school students (Grades 10-12).

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