How can I prove that $\frac{\mathbb R^n-\{0\} }{ \mathbb Z_2}$ is homotopic to $\frac{S^{n-1} }{ \mathbb Z_2}$

Question

I know that if $X=\mathbb R^n-\{0\}$ deformation retracts to $S^{n-1}$ Using the homotopy:

$$H:X\times [0,1] \longrightarrow X, \quad H(x,t)=tx+(1-t)\frac{x}{\lVert x \rVert}$$

And I can visually see that the identification points just move to the sphere and stay diametrically opposed. But I cannot find and homotopy that satisfied this.

I thought about $H(-x,t)=-H(x,t)$ but again I'm not sure