`(2(4k+1))/(3)ge(k(6+5)-3)/(2)`
Which of the following correctly describes the possible values of k in the inequality above?

Options

A. `kle-1`
B. `kge-1`
C. `kle(13)/(17)`
D. `kge(13)/(17)`

Explanation

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

### Step 1: Eliminate the fractions
To eliminate the fractions, we will multiply both sides of the inequality by the least common multiple (LCM) of the denominators, which is 6.

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

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

- Left side:
\[ 6 \cdot \frac{2(4k+1)}{3} = 2 \cdot 2(4k+1) = 4(4k+1) = 16k + 4 \]

- Right side:
\[ 6 \cdot \frac{k(6+5)-3}{2} = 3(k(11)-3) = 3(11k - 3) = 33k - 9 \]

So the inequality now looks like:
\[ 16k + 4 \geq 33k - 9 \]

### Step 3: Rearrange the inequality
Next, we will rearrange the inequality to isolate \( k \). We will move all terms involving \( k \) to one side and constant terms to the other side.

\[ 16k + 4 - 33k + 9 \geq 0 \]

This simplifies to:
\[ -17k + 13 \geq 0 \]

### Step 4: Isolate \( k \)
Now, isolate \( k \) by moving the constant to the other side:

\[ -17k \geq -13 \]

### Step 5: Divide by -17
When dividing by a negative number, we must reverse the inequality sign:

\[ k \leq \frac{13}{17} \]

### Final Answer
Thus, the solution to the inequality is:
\[ k \leq \frac{13}{17} \]