Impact on Interquartile range

The interquartile range measures the spread of the middle half of a dataset: $\text{IQR} = Q_3 - Q_1$.
On a box plot it is the width of the box, from the first quartile to the third.

When you add or replace a value, the quartiles can shift.
Suppose a new data point replaces the highest value in the set; the median, $Q_1$, and $Q_3$ may all move to new positions.
Where exactly they land depends on the other values in the data.

The important point is that $Q_1$ and $Q_3$ need not move by the same amount.
$Q_1$ might rise by $1$ while $Q_3$ rises by $5$, because each quartile is set by a different part of the data.
Since the two ends move differently, the gap between them — the IQR — changes.

So never assume a change leaves the IQR fixed; check both quartiles.
Find $Q_1$ and $Q_3$ before and after the change, then compare $Q_3 - Q_1$.
Only if both quartiles shift by the same amount, or neither moves, does the IQR stay the same.

A dataset has $Q_1 = 3$ and $Q_3 = 9$. What is its interquartile range?
The IQR is $Q_3 - Q_1$.
So the IQR is $9 - 3 = 6$.

After a new value is added, $Q_1$ rises from $3$ to $4$ but $Q_3$ stays at $9$. Does the IQR change?
The new IQR is $9 - 4 = 5$, compared with the old $9 - 3 = 6$.
So yes, the IQR changes, dropping from $6$ to $5$.

A change shifts both $Q_1$ and $Q_3$ up by exactly $2$. Does the IQR change?
If both quartiles move by the same amount, their difference is unchanged.
So the IQR stays the same.