Two conditions in a slice

[IvanBulkins]

Setting up the display of rows in Slice
There is a setting for displaying rows, its formula is 

AND(MONTH([Next_Date_Check])=MONTH(TODAY()),YEAR([Next_Date_Check])=Year(Today()))

We need to add another verification condition to this formula.
If in the Check table, the CheckPassed column is TRUE, then this row from SLICE is hidden.

This is how I write TRUE to the TestPassed column:
There is a virtual string Hidden with a formula

IF(IN([WorkerID], SELECT(Check[WorkerID], MONTH([Next_Date_Check]) = MONTH(TODAY()))), TRUE, FALSE)

There is also a Done button in Behavior with the following settings:
Action name Done
For a record of this table Check
Do this Data: set the values of some columns in this row
Set these columns TestPassed= Hidden

It turns out that when I click on a person in Slice, TRUE is written in the Check table in the CheckPassed row, and i want to hide this person from Slice if row is TRUE.

[lynchk21]

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It turns out that when I click on a person in Slice, TRUE is written in the Check table in the CheckPassed row, and i want to hide this person from Slice if row is TRUE.

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If you want to filter out records from your table slice when [CheckPassed]=TRUE, then you just need to and the condition [CheckPassed]<>TRUE to your slice condition

AND(MONTH([Next_Date_Check])=MONTH(TODAY()),YEAR([Next_Date_Check])=Year(Today()),[CheckPassed]<>TRUE)

. Not sure if I understood your issue correctly