A Quick Algebra Problem from the 2022 Oxford Maths Admission Test

How many solutions does this equation have?

This was the first question on the 2022 Mathematics Admissions Test, an entry test for students applying for a Mathematics or Computer Science Degree at Oxford University, Imperial College London and/or University of Warwick.

The question is multiple choice:

Question retyped by author

I found it interesting that the paper even provided a definition of the modulus or absolute value function |x|. I would have thought that any students applying for a Maths or Computer Science degree (especially at Oxford!) would have seen the function in high school, but perhaps this is not the case?

The graph of y = |x| is shown below (although the graph was not provided with the question, just the verbal definition).

The Modulus / Absolute Value function (source: Wikipedia commons)

Now, how can we count the number of solutions?

You may be able to think of a shortcut to count the solutions without actually…