If `6 ge x ge -2` and `4 ge y ge -4`, find the limits for `(y)/(x)`, where x and y are non-zero integers.

Options

A. `(y)/(x) ge 2, (y)/(x) le (2)/(3)`
B. `(y)/(x) ge (-2)/(3), (y)/(x) le 2`
C. `(y)/(x) ge (-2)/(3), (y)/(x) le (1)/(4)`
D. `(y)/(x) ge -4, (y)/(x) le 4`

Answer

To find the limits for \(\frac{y}{x}\) given the inequalities \(6 \geq x \geq -2\) and \(4 \geq y \geq -4\), we will analyze the possible values for \(x\) and \(y\) and compute the limits for the ratio \(\frac{y}{x}\).

### Step 1: Identify the ranges for \(x\) and \(y\)
- The values of \(x\) can range from \(-2\) to \(6\) (inclusive).
- The values of \(y\) can range from \(-4\) to \(4\) (inclusive).

### Step 2: Consider the non-zero integer constraints
Since \(x\) and \(y\) must be non-zero integers, we can list the possible values:
- Possible values for \(x\): \(-1, 1, 2, 3, 4, 5, 6\)
- Possible values for \(y\): \(-3, -2, -1, 1, 2, 3, 4\)

### Step 3: Calculate the limits for \(\frac{y}{x}\)
We will calculate \(\frac{y}{x}\) for the extreme values of \(y\) and \(x\).

1. **Maximum value of \(\frac{y}{x}\)**:
- To maximize \(\frac{y}{x}\), we need the maximum \(y\) and the minimum positive \(x\).
- Maximum \(y = 4\) and minimum positive \(x = 1\):
\[ \frac{y}{x} = \frac{4}{1} = 4 \]

2. **Minimum value of \(\frac{y}{x}\)**:
- To minimize \(\frac{y}{x}\), we need the minimum \(y\) and the maximum positive \(x\).
- Minimum \(y = -4\) and maximum \(x = 6\):
\[ \frac{y}{x} = \frac{-4}{6} = -\frac{2}{3} \]

3. **Consider negative values of \(x\)**:
- If \(x = -1\) and \(y = 4\):
\[ \frac{y}{x} = \frac{4}{-1} = -4 \]
- If \(x = -2\) and \(y = 4\):
\[ \frac{y}{x} = \frac{4}{-2} = -2 \]
- If \(x = -1\) and \(y = -4\):
\[ \frac{y}{x} = \frac{-4}{-1} = 4 \]

### Step 4: Compile the results
From the calculations:
- The maximum value of \(\frac{y}{x}\) is \(4\).
- The minimum value of \(\frac{y}{x}\) is \(-4\).

Thus, the limits for \(\frac{y}{x}\) are:
\[ -4 \leq \frac{y}{x} \leq 4 \]

### Final Answer:
The limits for \(\frac{y}{x}\) are \(-4 \leq \frac{y}{x} \leq 4\).