The expression you provided is:

\[
2|X| - \frac{|X| + X}{3}
\]

### Steps to simplify:

1. **Absolute values:** Since the absolute value \(|X|\) depends on the value of \(X\), we have to consider two cases:
   - Case 1: \(X \geq 0\)
   - Case 2: \(X < 0\)

---

#### **Case 1: \(X \geq 0\)**

In this case, \(|X| = X\). So, the expression becomes:

\[
2X - \frac{X + X}{3} = 2X - \frac{2X}{3}
\]

Simplifying:

\[
2X - \frac{2X}{3} = \frac{6X}{3} - \frac{2X}{3} = \frac{4X}{3}
\]

---

#### **Case 2: \(X < 0\)**

In this case, \(|X| = -X\). So, the expression becomes:

\[
2(-X) - \frac{-X + X}{3} = -2X - \frac{0}{3} = -2X
\]

---

### Final Simplified Form:

- When \(X \geq 0\), the expression simplifies to: \(\frac{4X}{3}\)
- When \(X < 0\), the expression simplifies to: \(-2X\)

### Piecewise Answer:

\[
f(X) =
\begin{cases}
\frac{4X}{3} & \text{if } X \geq 0 \\
-2X & \text{if } X < 0
\end{cases}
\]

---

Let me know if you want further details on this!

#### Related Questions:
1. How do you simplify other piecewise functions?
2. What is the significance of absolute values in equations?
3. How can absolute value affect the derivative of a function?
4. What are the properties of absolute value and how do they apply in algebraic expressions?
5. Can you graph this piecewise function?

**Tip:** Always break down absolute value functions into cases based on the sign of the variable for a clear solution.

Simplify the expression 2|X| - (|X| + X)/3 using piecewise function breakdown. Learn how absolute values affect algebraic expressions and their simplification in different cases, covering the core concepts of algebra and piecewise functions. Suitable for students in Grades 10-12.

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