The intercepts on x-axis made by tangents to the curve, y = ∫₀ˣ|t|dt, x∈R, which are parallel to the line y = 2x, are equal to

dy/dx = |x| = 2 ⇒ x = ±2 ⇒ y = ∫₀²|t|dt = 2 for x = 2 and y = ∫₋₂⁰|t|dt = 2 for x = -2
Hence the equations of the tangents are y - 2 = 2(x - 2) ⇒ y = 2x - 2 and y + 2 = 2(x + 2) ⇒ y = 2x + 2
Putting y = 0, we get x = 1 and -1.