If `(-3)/(x)+4le-11 and xgt0`, what ist he greatest possible value for x?

Answer

To solve the inequality \(-\frac{3}{x} + 4 \leq -11\) with the condition \(x > 0\), we will follow these steps:

### Step 1: Isolate the fraction
We start by isolating the term with \(x\):
\[ -\frac{3}{x} + 4 \leq -11 \]
Subtract 4 from both sides:
\[ -\frac{3}{x} \leq -11 - 4 \]
\[ -\frac{3}{x} \leq -15 \]

### Step 2: Multiply by -1
Next, we multiply both sides of the inequality by -1. Remember that when we multiply or divide an inequality by a negative number, we must reverse the inequality sign:
\[ \frac{3}{x} \geq 15 \]

### Step 3: Cross-multiply
Now we can cross-multiply to eliminate the fraction. Since \(x > 0\), we can do this without changing the inequality:
\[ 3 \geq 15x \]

### Step 4: Solve for \(x\)
Now, we will solve for \(x\) by dividing both sides by 15:
\[ \frac{3}{15} \geq x \]
This simplifies to:
\[ \frac{1}{5} \geq x \]
or
\[ x \leq \frac{1}{5} \]

### Step 5: Determine the greatest possible value of \(x\)
Given the condition \(x > 0\) and the result \(x \leq \frac{1}{5}\), the greatest possible value for \(x\) is:
\[ x = \frac{1}{5} \]

### Final Answer
The greatest possible value for \(x\) is \(\frac{1}{5}\).

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