In this question, the first thing to pay attention is whether air is pumped in or out of the system.
When air is pumped in, there is a force pushing down on the right side. This pushes water up on the left side ad well. The pressure of the water surface on the right side, caused by atmospheric pressure + pressure of air pumped in = water pressure from the excess amount of water on the left side + atmospheric pressure.
When air is pumped out, it creates somewhat of a vacuum. This allows the atmospheric pressure which is greater in intensity to push the water down from the left, raising the water level on the right.
Even though there is a cap on the left side as well, the air pressure from having air pumped in would result in pushing water on the right down and pushing water on the left to go up. This would compress the air on the left side instead of having equal water surface level. On both sides of the U-tube, water levels of identical depth would have identical pressure.
As mentioned earlier, when air is pumped out, the weight of the air pressing down on the right would be lesser, as such the weight of atmospheric air would be greater and push water from the left side downwards, pushing the right side upwards.
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