The aim of this research is to investigate how well high-achieving students entering tertiary-level education in Ireland understand school algebra. As part of a larger project, a 31-item test was developed to assess first-year undergraduate students' understanding of basic algebraic concepts. The test was administered online to students studying at least one mathematics module at tertiary level and received 327 responses. In this article, we study how the subset of high-achieving undergraduates in our sample performed on the test. The results demonstrated a very high level of understanding among students, as befits their level of study and prior achievement relative to the difficulty of the test. However, one subsection of the test stood out as being disproportionately difficult for these students. The section focused on valid solutions of equations and inequalities. The items in question are described in detail in this article as is the associated data. Our analysis shows that this topic is an area of concern even for high-achieving undergraduates and so deserves further attention. We conclude with a discussion of the implications of this research and details of the larger project.