If `1/2 (3/5 x+4) ge 1/3 (x-6) , x in R` , then

Options

A. (a) `x in ( -oo ,120]`
B. (b) `x in [120,oo)`
C. (c) `x in (- oo,120)`
D. (d) `x in (120,oo)`

Answer

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

### Step 1: Eliminate the fractions
Multiply both sides of the inequality by 6 (the least common multiple of 2 and 3) to eliminate the fractions:

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

This simplifies to:

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

### Step 2: Distribute the constants
Distributing on both sides gives:

\[ \frac{9}{5} x + 12 \geq 2x - 12 \]

### Step 3: Rearrange the inequality
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:

\[ \frac{9}{5} x - 2x \geq -12 - 12 \]

This can be rewritten as:

\[ \frac{9}{5} x - \frac{10}{5} x \geq -24 \]

### Step 4: Combine like terms
Combining the \(x\) terms:

\[ -\frac{1}{5} x \geq -24 \]

### Step 5: Multiply by -5
Since we are multiplying by a negative number, we must reverse the inequality sign:

\[ x \leq 120 \]

### Step 6: Write the solution in interval notation
The solution indicates that \(x\) can take any value less than or equal to 120. In interval notation, this is expressed as:

\[ x \in (-\infty, 120] \]

### Conclusion
Thus, the correct option is:

**Option A: \( x \in (-\infty, 120] \)**

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