Solve the equation:
`(4+2x)/(3)ge(x)/(2)-3`.

Answer

To solve the inequality \(\frac{4 + 2x}{3} \geq \frac{x}{2} - 3\), we will follow these steps:

### Step 1: Eliminate the fractions
To eliminate the fractions, we will find a common denominator. The least common multiple (LCM) of 3 and 2 is 6. We will multiply both sides of the inequality by 6:

\[ 6 \cdot \frac{4 + 2x}{3} \geq 6 \cdot \left(\frac{x}{2} - 3\right) \]

### Step 2: Simplify both sides
Now, simplify both sides of the inequality:

\[ 2(4 + 2x) \geq 3\left(x - 6\right) \]

### Step 3: Distribute
Distributing on both sides gives:

\[ 8 + 4x \geq 3x - 18 \]

### Step 4: Rearrange the inequality
Next, we will move all terms involving \(x\) to one side and constant terms to the other side. Subtract \(3x\) from both sides:

\[ 8 + 4x - 3x \geq -18 \]

This simplifies to:

\[ 8 + x \geq -18 \]

### Step 5: Isolate \(x\)
Now, subtract 8 from both sides to isolate \(x\):

\[ x \geq -18 - 8 \]

This simplifies to:

\[ x \geq -26 \]

### Step 6: Write the solution set
The solution can be expressed in interval notation as:

\[ x \in [-26, \infty) \]

### Summary of the solution:
The solution to the inequality \(\frac{4 + 2x}{3} \geq \frac{x}{2} - 3\) is:

\[ x \geq -26 \]

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