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.

Question

UpStudy AI · Accepted Answer

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.
$$ y \geq 2 $$ On the number line, this is represented by a closed circle at 2 and an arrow pointing to the right, indicating all values greater than or equal to 2 satisfy the inequality.

Solve the inequality
and illuatrate your anewer on a number line.

Solución de inteligencia artificial de Upstudy

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Solución

Step 1: Simplify Both Sides

Step 2: Collect Like Terms

Solution

Number Line Illustration

Beyond the Answer

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Solve the inequality and illuatrate your anewer on a number line.