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 provided curves depict absolute value equations that define a symmetrical diamond-shaped region. The equations of the lines are \( y - 1 = |x| \) and \( y + 1 = |x| \). These can be expressed as \( y = |x| + 1 \) and \( y = -|x| - 1 \). Collectively, these lines form a rhombus centered at the origin.

Step 1: Determine intersection points The given equations intersect at four vertices: - Upper vertex: \( (0,1) \) - Lower vertex: \( (0,-1) \) - Right vertex: \( (1,0) \) - Left vertex: \( (-1,0) \) 

Step 2: Calculate the area The total enclosed area is the sum of four congruent right triangles, one in each quadrant. The area of a single triangle is calculated as: \[ A_{\triangle} = \frac{1}{2} \times OA \times OC = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}. \] With four such triangles, the total enclosed area is: \[ A = 4 \times \frac{1}{2} = 2. \] 

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