The core idea is to find the value of $k_1$ that satisfies the following inequality: \begin{equation} k_1\vert x\vert + k_2 > \vert x + k_3\vert \tag{1} \end{equation} where $k_2 > \vert k3 \vert > 0$, $k_1 \geq 0$ and $x \in \mathbb{R}$. I have tried to approach this problem using triangle inequality as follows:
CASE 1: \begin{align} k_1\vert x \vert + \vert k_3 \vert &> \vert x \vert + k_2 > \vert x + k_3 \vert \tag{2}\\ \Rightarrow k_1 &> \frac{\vert k_3 \vert - k_2}{\vert x \vert} + 1 \tag{3} \end{align}
CASE 2: \begin{align} \vert x \vert + \vert k_3 \vert > k_1\vert x \vert + k_2 > \vert x + k_2 \vert \end{align}
- The left side inequality gives $\frac{\vert k_2 \vert - k_3}{\vert x \vert} + 1 > k_1$.
- However, I get stuck with the upper limit inequality. How do I proceed with this calculation?
