I have some problems to make from primal problem to dual problem.

If I have the primal problem like down below.

Problem

$$\text{Min} \qquad 3w_1 + 4w_2 + 5w_3$$ $$w_1 - w_2 \le ε_1$$ $$w_2 - w_3 \le ε_2$$ $$w_3 \le ε_3$$ $$1 \le 2ε_1 +3ε_2 + 4ε_3$$ $$w_1 + w_2 + w_3 = 1$$

then I think its dual problem with dual variable x is

$$\text{Max} \qquad ε_1x_1 + ε_2x_2 + ε_3x_3 + (2ε_1 +3ε_2 + 4ε_3)x_4 + x_5$$ $$s.t. x_1 + x_5 \le3$$ $$x_2 \le 4$$ $$x_3 \le 5$$ $$\text{where} \quad r_4 \quad \text{and} \quad r_5 \quad \text{are} \quad \text{unrestricted} \quad \text{in } \quad \text{sign}.$$

am I right?