Solve system of inequations with absolute values

Question

$$ \left\{ \begin{array}{c} |c_1-3c_2|<\delta \\ |c_1+c_2|<\delta \end{array} \right. $$

Could you provide how $|c_1|$ and $|c_2|$ are limited by $\delta$ in such a form: $|c_i| < k\delta $.

Thank you.

Answer 1

Consider on plane $\mathbb{R}^2$ the four lines: $$ x - 3y = \delta\\ x - 3y = -\delta\\ x + y = \delta\\ x + y = -\delta $$ They form a parallelogram. The region you describe is the interior of this parallelogram. If you make a drawing, you will see clearly that $$ |c_1| <\delta \quad \text{and} \quad |c_2| < \delta/2. $$