solve the inequalities for real x `(1)/(2) ((3x)/(5) + 4 )ge (1)/(3) (x - 6)`

Explanation

To solve the inequality \(\frac{1}{2} \left( \frac{3x}{5} + 4 \right) \geq \frac{1}{3} (x - 6)\), we can follow these steps:

### Step 1: Eliminate the fractions
To eliminate the fractions, we can multiply both sides of the inequality by the least common multiple (LCM) of the denominators, which is 30.

\[ 30 \cdot \left( \frac{1}{2} \left( \frac{3x}{5} + 4 \right) \right) \geq 30 \cdot \left( \frac{1}{3} (x - 6) \right) \]

### Step 2: Simplify both sides
Now, we simplify both sides:

\[ 15 \left( \frac{3x}{5} + 4 \right) \geq 10(x - 6) \]

Distributing on both sides:

\[ 15 \cdot \frac{3x}{5} + 15 \cdot 4 \geq 10x - 60 \]

This simplifies to:

\[ 9x + 60 \geq 10x - 60 \]

### Step 3: Rearrange the inequality
Next, we rearrange the inequality to isolate \(x\):

\[ 9x + 60 + 60 \geq 10x \]

This simplifies to:

\[ 9x + 120 \geq 10x \]

Now, subtract \(9x\) from both sides:

\[ 120 \geq 10x - 9x \]

This simplifies to:

\[ 120 \geq x \]

### Step 4: Write the final solution
Thus, we can write the solution as:

\[ x \leq 120 \]