Question 1178643
This is a challenging problem that requires geometric reasoning and careful consideration of the function graphs. Let's break down the proof:

**1. Understand the Graphs:**

* **f(x) = 1 - |x - 1|:**
    * This is an absolute value function, creating a "V" shape.
    * The vertex of the "V" is at (1, 1).
    * The graph intersects the x-axis at x = 0 and x = 2.
    * The graph is symmetric about the line x = 1.
* **g(x) = 2x - a:**
    * This is a linear function with a slope of 2.
    * The y-intercept is -a.
    * Since a ∈ (1, 2), the y-intercept is between -2 and -1.

**2. Visualize the Bounded Area:**

* The graphs of f(x) and g(x) will intersect at two points, creating a bounded area.
* The shape of this bounded area will be a triangle.

**3. Find the Intersection Points:**

* To find the intersection points, set f(x) = g(x):
    * 1 - |x - 1| = 2x - a
* We need to consider two cases for the absolute value:
    * **Case 1: x ≥ 1**
        * 1 - (x - 1) = 2x - a
        * 2 - x = 2x - a
        * 3x = 2 + a
        * x = (2 + a) / 3
    * **Case 2: x < 1**
        * 1 - (1-x) = 2x -a
        * x = 2x -a
        * x = a
* Because a is within the interval (1,2) then the intersection points are x = a and x = (2+a)/3.

**4. Geometric Approach:**

* **Triangle Formation:** The bounded area is a triangle.
* **Base of the Triangle:** The base of the triangle is the distance between the two intersection points:
    * Base = |(2 + a) / 3 - a| = |(2 - 2a) / 3| = (2 - 2a) / 3 (since a<2).
* **Height of the Triangle:**
    * The height is the vertical distance from the vertex of f(x) (1, 1) to the line g(x).
    * The x-coordinate of the vertex of f(x) is x=1.
    * The y-coordinate of g(x) at x=1 is g(1)=2-a.
    * The Height is 1-(2-a) = a-1
* **Area of the Triangle:**
    * Area = (1/2) * Base * Height
    * Area = (1/2) * [(2 - 2a) / 3] * (a - 1)
    * Area = (1/6) * (2 - 2a) * (a - 1)
    * Area = (-1/3) * (a - 1) * (a - 1)
    * Area = (-1/3) * (a - 1)^2
    * Area = (1/3) * (1-a)^2

**5. Prove Area < 1/3:**

* Since a ∈ (1, 2), (1 - a) is a negative value.
* Therefore (1-a)^2 is a positive value.
* Since a is between 1 and 2 then 0 < (a-1) < 1.
* Therefore 0 < (1-a)^2 < 1.
* Therefore 0 < (1/3)*(1-a)^2 < 1/3.
* Therefore the area is less than 1/3.

**Conclusion:**

For any value of 'a' in the interval (1, 2), the area bounded by the graphs of f(x) and g(x) is less than 1/3. This is proven using geometric properties of the graphs and without the use of differentiation or integration.