Question 1172991
Let's break down this problem step-by-step.

**a. Mathematical Statements (Inequalities)**

* **Variables:**
    * Let 'x' be the amount deposited in the 4% interest bank.
    * Let 'y' be the amount deposited in the 6% interest bank.

* **Inequalities:**
    1.  **Total Deposit:** The total deposit is at least Php 150,000.
        * x + y ≥ 150,000
    2.  **Total Interest:** The total interest is at most Php 12,000.
        * 0.04x + 0.06y ≤ 12,000
    3.  **Non-negativity:** Deposits cannot be negative.
        * x ≥ 0
        * y ≥ 0

**b. Graph the System**

1.  **x + y ≥ 150,000**
    * Graph the line x + y = 150,000.
    * Intercepts: (150,000, 0) and (0, 150,000).
    * Shade the region above the line.

2.  **0.04x + 0.06y ≤ 12,000**
    * Simplify: Multiply by 100: 4x + 6y ≤ 1,200,000.
    * Simplify further: Divide by 2: 2x + 3y ≤ 600,000.
    * Graph the line 2x + 3y = 600,000.
    * Intercepts: (300,000, 0) and (0, 200,000).
    * Shade the region below the line.

3.  **x ≥ 0 and y ≥ 0**
    * This restricts the solution to the first quadrant.

4.  **The Solution Region:** The solution is the area where all shaded regions overlap.

**c. Two Possible Amounts**

1.  **Point 1: (150,000, 0)**
    * x = 150,000 (4% bank)
    * y = 0 (6% bank)
    * Total deposit: 150,000
    * Total interest: 0.04(150,000) + 0.06(0) = 6,000
    * This point satisfies both conditions.

2.  **Point 2: (100,000, 200,000)**
    * x=100,000 (4% bank)
    * y=200,000 (6% bank)
    * Total deposit: 300,000
    * Total interest: 0.04(100,000) + 0.06(200,000) = 4,000 + 12,000 = 16,000. This point does not satisfy the total interest constraint.
    * We need a point that is within the solution set.
    * Let's test the point (200,000, 100,000)
    * x=200,000 (4% bank)
    * y=100,000 (6% bank)
    * Total deposit: 300,000
    * Total interest: 0.04(200,000) + 0.06(100,000) = 8,000 + 6,000 = 14,000. This point does not satisfy the total interest constraint.
    * Let's test the point (250,000, 50,000)
    * x=250,000 (4% bank)
    * y=50,000 (6% bank)
    * Total deposit: 300,000
    * Total interest: 0.04(250,000) + 0.06(50,000) = 10,000 + 3,000 = 13,000. This point does not satisfy the total interest constraint.
    * Let's test the point (180,000, 20,000)
    * x=180,000 (4% bank)
    * y=20,000 (6% bank)
    * Total deposit: 200,000
    * Total interest: 0.04(180,000) + 0.06(20,000) = 7,200 + 1,200 = 8,400. This point satisfies the constraints.

**d. Bank Account Preference**

* If I were Mr. Agoncillo, I would place a greater amount of money in the 6% interest bank.
* **Reasoning:**
    * The 6% interest bank provides a higher return on investment.
    * Even though there is a constraint on the total interest, maximizing the amount in the 6% bank within the constraints will yield a higher overall return.
    * However, if Mr. Agoncillo has other considerations such as risk tolerance or liquidity needs, he might opt for a different allocation.