Question

The area bounded by \( y - 1 = |x| \) and \( y + 1 = |x| \) is: (a) \( \frac{1}{2} \)

• A. 1/2
• B. 1
• C. 2
• D. 0

Hint:

For absolute value equations forming enclosed regions, visualize their graphs as V-shaped or X-shaped intersections. The enclosed area can often be computed using basic geometry.

Answer 1

The Correct Option is C

The given curves represent absolute value equations that form a symmetric diamond-shaped region. The equations of the lines are: \[ y - 1 = |x| \quad \text{and} \quad y + 1 = |x| \] which can be rewritten as four separate linear equations: \[ y = |x| + 1 \quad \text{and} \quad y = -|x| - 1. \] These lines form a rhombus centered at the origin. 

Step 1: Find the intersection points The given equations intersect at four points: - Top vertex: \( (0,1) \) - Bottom vertex: \( (0,-1) \) - Right vertex: \( (1,0) \) - Left vertex: \( (-1,0) \) 

Step 2: Compute the area The total enclosed area is the sum of four identical right triangles (one in each quadrant). The area of one such triangle is: \[ A_{\triangle} = \frac{1}{2} \times OA \times OC = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}. \] Since there are four such triangles, the total enclosed area is: \[ A = 4 \times \frac{1}{2} = 2. \] 

Final Answer: The total bounded area is \( 2 \).