1. What is Ling-Ling not good at?

2. What isn't Ling-Ling good at?

3. Why don't you like apples?
4. Why do you not like apples?

5. Why can't you solve the problem?
6. Why can you not solve the problem?
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Are they all grammatical? Which expressions are commonly used?

All are correct ... and all are used interchangeably!

Yes, all of the provided sentences are grammatically correct. However, some expressions are more commonly used than others. Let's break down each sentence and discuss their usage:

1. What is Ling-Ling not good at? - This is a commonly used sentence structure to inquire about Ling-Ling's abilities or skills.
2. What isn't Ling-Ling good at? - This sentence structure is less frequently used, but still grammatically correct. It emphasizes the negative aspect of Ling-Ling's abilities.

3. Why don't you like apples? - This is a commonly used sentence to ask someone why they have a dislike for apples.
4. Why do you not like apples? - This sentence structure is less commonly used, but still grammatically correct. It presents a formal or slightly less common way of asking the same question.

5. Why can't you solve the problem? - This is a commonly used sentence structure to inquire about why someone is unable to solve a problem.
6. Why can you not solve the problem? - This sentence structure is less frequently used but still grammatically correct. It presents a formal or slightly less common way of asking the same question.

In summary, all of the provided sentences are grammatically correct, but some expressions are more commonly used in everyday conversation or writing.