Quasilinear preference is defined to be:

$x \sim y \Rightarrow x+ae_1 \sim y+ae_1$ and $x + ae_1 \succ x$ with $e_1 = (1,0,0,...)$,

Given a quasilinear preference, if f $x \succeq y - ae_1$, does it mean $x + ae_1 \succeq y$?

I know that $x + ae_1 \succ x$ but how do I compare $x+ae_1$ with $y$? Drawing a diagram makes me see it but I can't prove it rigorously.

EDIT: Included the definition for quasilinear preference.

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    $\begingroup$ Could you please define what you mean by "quasilinear preference". Doing so may help you answer your own question. $\endgroup$ – Giskard Sep 26 '19 at 5:26
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    $\begingroup$ Now that you have included your definition, are you sure you do not know how to proceed? There is no trick at all, you just apply the definition. $\endgroup$ – Giskard Sep 26 '19 at 21:35
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    $\begingroup$ Could you elaborate on it? $\endgroup$ – Rainroad Sep 26 '19 at 23:15
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    $\begingroup$ If the OP had used the utility function definition of quasilinear preferences, the question would be easy. However, the OP is using a different definition which makes the problem more difficult and more interesting. $\endgroup$ – Angela Pretorius Sep 27 '19 at 7:15
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    $\begingroup$ Hint: Let $z=y-a\mathbf e_1$, so that $x\succsim y-a\mathbf e_1$ becomes $x\succsim z$. Apply the definition of quasilinear preference. $\endgroup$ – Herr K. Sep 27 '19 at 19:21

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