Solve : `3((3x)/(5) + 4) ge 2(x-6)`

Explanation

To solve the inequality \( 3\left(\frac{3x}{5} + 4\right) \geq 2(x - 6) \), we will follow these steps:

### Step 1: Distribute the 3 on the left side
We start by distributing the 3 into the expression inside the parentheses on the left side:
\[ 3\left(\frac{3x}{5}\right) + 3 \cdot 4 \geq 2(x - 6) \]
This simplifies to:
\[ \frac{9x}{5} + 12 \geq 2(x - 6) \]

### Step 2: Distribute the 2 on the right side
Next, we distribute the 2 on the right side:
\[ \frac{9x}{5} + 12 \geq 2x - 12 \]

### Step 3: Move all terms involving \(x\) to one side and constant terms to the other side
Now, we will move \(2x\) to the left side and \(12\) to the right side:
\[ \frac{9x}{5} - 2x \geq -12 - 12 \]
This simplifies to:
\[ \frac{9x}{5} - 2x \geq -24 \]

### Step 4: Combine like terms
To combine the terms involving \(x\), we need a common denominator. The LCM of 5 and 1 is 5:
\[ \frac{9x}{5} - \frac{10x}{5} \geq -24 \]
This simplifies to:
\[ \frac{-x}{5} \geq -24 \]

### Step 5: Multiply both sides by -5
Since we are multiplying by a negative number, we must reverse the inequality:
\[ -x \leq -120 \]
This simplifies to:
\[ x \geq 120 \]

### Step 6: Write the solution in interval notation
The solution can be expressed in interval notation as:
\[ x \in [120, \infty) \]

### Final Answer
Thus, the solution to the inequality \( 3\left(\frac{3x}{5} + 4\right) \geq 2(x - 6) \) is:
\[ x \geq 120 \]

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