zur Neige gehen
to be running out
To be almost used up; to be close to the end. Used for supplies, resources, or also time.
Unsere Geduld geht langsam zur Neige.
Our patience is slowly running out.

end / incline
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to be running out
To be almost used up; to be close to the end. Used for supplies, resources, or also time.
Unsere Geduld geht langsam zur Neige.
Our patience is slowly running out.

The noun 'die Neige' is related to the verb 'neigen' (to tilt / to bend), which means 'kippen' (to tip) or 'sich beugen' (to bend). The phrase 'zur Neige gehen' (to be running out) describes figuratively how the contents of a bottle come to an end when you tilt it.
Er neigte die Flasche, um den letzten Rest Wein einzuschenken.
He tilted the bottle to pour the last bit of wine.
In modern German, 'Neige' is used almost exclusively in the fixed expression 'zur Neige gehen' (to be running out) or 'sich dem Ende zuneigen' (to be drawing to a close). The word hardly ever appears on its own anymore.
Der Winter geht langsam zur Neige und der Frühling beginnt.
Winter is slowly drawing to a close, and spring is beginning.
The phrase 'zur Neige gehen' (to be running out) is often used for abstract things like 'Geduld' (patience), 'Zeit' (time), or 'Kraft' (strength), but also for concrete supplies like 'Geld' (money), 'Essen' (food), or 'Wein' (wine).
Nach dem langen Marsch gingen seine Kräfte zur Neige.
After the long march, his strength was running out.
The meaning 'Neigung' (inclination) or 'Senkung' (lowering) is outdated and is used only in very elevated or literary language. You should not use this meaning in everyday speech.
In dem historischen Roman erwiderte sie seinen Gruß mit einer leichten Neige des Hauptes.
In the historical novel, she returned his greeting with a slight inclination of her head.
end
'das Ende' and 'die Neige' both have to do with a finish or with only a small amount left. 'das Ende' names the final point directly: something is over or stops. 'die Neige' is almost always used in the phrase 'zur Neige gehen' (to be running out). This means that something is almost used up, but not completely over yet. Typical examples are supplies, money, time, or strength. The confusion arises because 'zur Neige gehen' means: it is approaching the end. If the end has already been reached, you say 'Ende'. If only a little is left, you say 'zur Neige gehen'.

Vor dem Wochenende gehen unsere Vorräte langsam zur Neige.
Before the weekend, our supplies are slowly running out.
Nach drei Stunden kam das Ende der Besprechung endlich näher.
After three hours, the end of the meeting finally drew closer.