If `(3(x-2))/(5)gt=(5(2-x))/(3)`, then x belongs to the interval

Options

A. `(2, oo)`
B. `[2, oo)`
C. `(-oo, 2]`
D. none of these

Answer

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

### Step 1: Cross Multiply
We start by cross-multiplying to eliminate the fractions. This gives us:
\[ 3(x - 2) \cdot 3 \geq 5(2 - x) \cdot 5 \]
This simplifies to:
\[ 9(x - 2) \geq 25(2 - x) \]

### Step 2: Expand Both Sides
Next, we expand both sides of the inequality:
\[ 9x - 18 \geq 50 - 25x \]

### Step 3: Combine Like Terms
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:
\[ 9x + 25x \geq 50 + 18 \]
This simplifies to:
\[ 34x \geq 68 \]

### Step 4: Solve for \(x\)
Now, we divide both sides by 34:
\[ x \geq 2 \]

### Step 5: Write the Interval
Since \(x\) is greater than or equal to 2, we can express the solution in interval notation:
\[ x \in [2, \infty) \]

### Final Answer
Thus, the solution to the inequality is:
\[ x \text{ belongs to the interval } [2, \infty) \]

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