2. Solve the inequality \( \frac{2}{3} y-\frac{1}{3}(1-2 y) \geq y+\left\lvert\, \frac{1}{3}\right. \) and illuatrate your anewer on a number line.

To solve the inequality: \[ \frac{2}{3} y - \frac{1}{3}(1 - 2y) \geq y + \left| \frac{1}{3} \right| \] follow these steps: ### Step 1: Simplify Both Sides 1. **Simplify the Absolute Value**: \[ \left| \frac{1}{3} \right| = \frac{1}{3} \] 2. **Expand the Left Side**: \[ \frac{2}{3} y - \frac{1}{3}(1 - 2y) = \frac{2}{3} y - \frac{1}{3} + \frac{2}{3} y = \frac{4}{3} y - \frac{1}{3} \] 3. **Write the Simplified Inequality**: \[ \frac{4}{3} y - \frac{1}{3} \geq y + \frac{1}{3} \] ### Step 2: Collect Like Terms 1. **Subtract \( y \) from Both Sides**: \[ \frac{4}{3} y - y - \frac{1}{3} \geq \frac{1}{3} \] 2. **Convert \( y \) to Thirds**: \[ \frac{4}{3} y - \frac{3}{3} y - \frac{1}{3} \geq \frac{1}{3} \] 3. **Combine Like Terms**: \[ \frac{1}{3} y - \frac{1}{3} \geq \frac{1}{3} \] 4. **Add \( \frac{1}{3} \) to Both Sides**: \[ \frac{1}{3} y \geq \frac{2}{3} \] 5. **Multiply Both Sides by 3**: \[ y \geq 2 \] ### Solution \[ y \geq 2 \] ### Number Line Illustration Below is the graph of the solution on a number line: \[ \begin{array}{c} \text{---|---|---|---|---|---|---|---} \\ \quad 1 \quad \quad 2 \quad \quad 3 \quad \end{array} \] - **Closed Circle at \( y = 2 \)**: Indicates that \( y = 2 \) is included in the solution. - **Shaded Region to the Right of 2**: Represents all values \( y \geq 2 \). \[ \begin{tikzpicture} \draw[<->] (-1,0) -- (5,0) node[right] {$y$}; \foreach \x in {0,2,4} \draw (\x,3pt) -- (\x,-3pt) node[below] {$\x$}; \draw[very thick] (2,0) circle (2pt); \draw[thick, ->] (2,0) -- (4.5,0); \end{tikzpicture} \] In this illustration: - The **closed dot** at \( y = 2 \) signifies \( y = 2 \) is part of the solution. - The **arrow** indicates that all values greater than or equal to 2 satisfy the inequality.