Step 1: Understanding the Concept:
This question tests Phrasal Verbs—combinations of a verb and a preposition/adverb that have a specific idiomatic meaning.
The sentence establishes a situation of conflict or discipline: Sharon is upset (crying/complaining), but the Mother remains firm in her stance.
We need a phrasal verb that means "to reduce pressure" or "to stop being strict."
Step 2: Detailed Explanation:
Analysis of the phrasal verb options:
1. Let up (Option D): When you "let up on" someone, you become less intense or stop being harsh with them.
This fits the context perfectly: despite the emotional outburst, the mother did not reduce the pressure of her discipline or demands.
This is the standard usage for situations involving persistent pressure, weather, or effort.
2. Let in (Option A): This means to allow someone or something to enter a physical space.
"Mom did not let in on her" is grammatically incorrect and makes no sense in this context.
3. Let off (Option B): To "let someone off" means to allow them to escape punishment or to give them a lighter punishment than they deserve.
While the meaning is similar to what we need, the preposition "on" makes it incorrect. You "let someone off" (no "on" required).
Example: "Mom did not let Sharon off." (Correct). "Mom did not let off on her." (Incorrect).
4. Let out (Option C): This means to release something (like a secret or a sound).
Example: "Sharon let out a cry."
It does not fit the blank as it doesn't describe the mother's behavior toward the daughter's discipline.
Therefore, "let up" is the only option that is both grammatically compatible with the preposition "on" and logically consistent with the sentence's meaning.
Step 3: Final Answer:
The correct phrasal verb to complete the sentence is "let up."