The intercepts on $x$ -axis made by tangents to the curve $y={\int }_0^x|t|dt$, $x\in \mathbb{R}$, which are parallel to the line $y=2x$, are equal to:

1. Curve: $y={\int }_0^x|t|dt$.2. $\frac{dy}{dx}=|x|$ (Leibniz Rule).3. Given tangent is parallel to $y=2x$, so $|x|=2\Rightarrow x=\pm 2$.4. Case 1: $x=2$. $y={\int }_0^{2}tdt=[\frac{t^2}{2}{]}_0^2=2$. Point $(2,2)$.Tangent: $y-2=2(x-2)\Rightarrow y=2x-2$. $x$ -intercept (put $y=0$): $x=1$.5. Case 2: $x=-2$. $y={\int }_0^{-2}-tdt=[-\frac{t^2}{2}{]}_0^{-2}=-2$. Point $(-2,-2)$.Tangent: $y+2=2(x+2)\Rightarrow y=2x+2$. $x$ -intercept: $x=-1$.Intercepts are $\pm 1$.