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To find the minimum value of \( z = x - 5y \) under the given constraints:

\[
\begin{cases}
4x - 3y - 3 \geq 0 \\
x - 2y - 2 \leq 0 \\
2x + 6y - 9 \leq 0 
\end{cases}
\]

First, let's rewrite these inequalities for clarity:

1. \( 4x - 3y \geq 3 \)
2. \( x - 2y \leq 2 \)
3. \( 2x + 6y \leq 9 \)

### Step-by-Step Solution

1. **Express y in terms of x (or vice versa) from each inequality:**

   From the first inequality:
   \[
   4x - 3y \geq 3 \implies y \leq \frac{4x - 3}{3}
   \]

   From the second inequality:
   \[
   x - 2y \leq 2 \implies y \geq \frac{x - 2}{2}
   \]

   From the third inequality:
   \[
   2x + 6y \leq 9 \implies y \leq \frac{9 - 2x}{6}
   \]

2. **Combine these inequalities to find the feasible region for (x, y):**

   We have:
   \[
   \frac{x - 2}{2} \leq y \leq \min \left( \frac{4x - 3}{3}, \frac{9 - 2x}{6} \right)
   \]

3. **Graphical Method:**

   To find the feasible region, we need to plot these lines and determine the intersection points:
   - \( y = \frac{4x - 3}{3} \)
   - \( y = \frac{x - 2}{2} \)
   - \( y = \frac{9 - 2x}{6} \)

4. **Finding Intersection Points:**

   - Intersection of \( y = \frac{4x - 3}{3} \) and \( y = \frac{x - 2}{2} \):
     \[
     \frac{4x - 3}{3} = \frac{x - 2}{2}
     \]
     Solving this equation for \( x \):
     \[
     8x - 6 = 3x - 6 \implies 5x = 0 \implies x = 0, \quad y = -1
     \]

   - Intersection of \( y = \frac{4x - 3}{3} \) and \( y = \frac{9 - 2x}{6} \):
     \[
     \frac{4x - 3}{3} = \frac{9 - 2x}{6}
     \]
     Solving this equation for \( x \):
     \[
     8x - 6 = 9 - 2x \implies 10x = 15 \implies x = 1.5, \quad y = 1
     \]

   - Intersection of \( y = \frac{x - 2}{2} \) and \( y = \frac{9 - 2x}{6} \):
     \[
     \frac{x - 2}{2} = \frac{9 - 2x}{6}
     \]
     Solving this equation for \( x \):
     \[
     3(x - 2) = 9 - 2x \implies 3x - 6 = 9 - 2x \implies 5x = 15 \implies x = 3, \quad y = 0.5
     \]

5. **Evaluate \( z = x - 5y \) at these intersection points:**

   - At \( (0, -1) \):
     \[
     z = 0 - 5(-1) = 5
     \]

   - At \( (1.5, 1) \):
     \[
     z = 1.5 - 5(1) = -3.5
     \]

   - At \( (3, 0.5) \):
     \[
     z = 3 - 5(0.5) = 3 - 2.5 = 0.5
     \]

The minimum value of \( z \) is \(-3.5\).

Thus, the minimum value of \( z = x - 5y \) is \( \boxed{-3.5} \).

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How to graph linear inequalities?
2. How to find the intersection points of lines algebraically?
3. How to solve systems of linear inequalities?
4. How to minimize or maximize a linear function subject to constraints?
5. What are the feasible regions in linear programming?

**Tip:** When solving systems of inequalities, visualizing the feasible region using a graph can greatly aid in understanding the problem and finding the solution.

Learn how to minimize the function z = x - 5y using linear programming constraints. Explore step-by-step solutions and graphical methods to find the minimum value of z under given inequalities. Suitable for advanced high school and college-level students.

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