If `(5-2xx)/(3)le(X)/(6)-5,` then `X in`

Options

A. `[2,oo)`
B. `[-8,8]`
C. `[4,oo)`
D. `[8,oo)`

Answer

To solve the inequality \( \frac{5 - 2x}{3} \leq \frac{x - 6}{-5} \), we will follow these steps:

### Step 1: Rewrite the Inequality
Start by rewriting the inequality for clarity:

\[ \frac{5 - 2x}{3} \leq \frac{x - 6}{-5} \]

### Step 2: Eliminate the Fractions
To eliminate the fractions, we can multiply both sides by \( -15 \) (the least common multiple of 3 and -5). Remember that multiplying by a negative number reverses the inequality sign:

\[ -15 \cdot \frac{5 - 2x}{3} \geq -15 \cdot \frac{x - 6}{-5} \]

This simplifies to:

\[ -5(5 - 2x) \geq 3(x - 6) \]

### Step 3: Distribute
Now, distribute on both sides:

\[ -25 + 10x \geq 3x - 18 \]

### Step 4: Collect Like Terms
Next, we will collect all \( x \) terms on one side and constant terms on the other side:

\[ 10x - 3x \geq -18 + 25 \]

This simplifies to:

\[ 7x \geq 7 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 7:

\[ x \geq 1 \]

### Final Answer
Thus, the solution to the inequality is:

\[ x \in [1, \infty) \]