Question

3. Determine o produto de todas as raízes da equação |5x – 1| = 1 – x

Answer 1

Condição de existência:

$\mathsf{1-x\ge0\times(-1)}\\\mathsf{x-1\le0\to~x\le1}$

$\mathsf{|5x-1|=1-x}\to\begin{cases}\mathsf{5x-1=1-x}\\\mathsf{5x-1=x-1}\end{cases}$

$\mathsf{5x-1=1-x}\\\mathsf{5x+x=1+1}\\\mathsf{6x=2}\\\mathsf{x=\dfrac{2\div2}{6\div2}}\\\mathsf{x=\dfrac{1}{3}}$

$\mathsf{5x-1=x-1}\\\mathsf{5x-x=1-1\to~4x=0}\\\large\boxed{\boxed{\boxed{\boxed{\mathsf{x=0}}}}}$

Portanto

$\huge\boxed{\boxed{\boxed{\boxed{\mathsf{s=\{0,\dfrac{1}{3}\}}}}}}$

O produto das raízes é

$\boxed{\boxed{\boxed{\boxed{\mathsf{0.\dfrac{1}{3}=0}}}}}$