Louis La Minh Hoang

This lesson on inequalities went far beyond the basic “solve and draw a number line” approach. We focused on hard inequalities that require careful reasoning, transformations, and a strong sense of structure rather than quick calculations. It felt much more like problem-solving than routine algebra.

We worked with polynomial and rational inequalities, where the key challenge was not solving equations, but understanding how the sign of an expression changes across different intervals. Instead of blindly moving terms around, we had to factor expressions completely, identify critical points, and analyze each interval one by one. A single mistake in sign analysis could flip the entire solution.

Some problems involved absolute value inequalities, where the expression had to be split into multiple cases. These were tricky because they forced us to think logically about what the inequality really means, not just apply a formula. Other questions combined inequalities with functions, requiring us to consider domains, undefined points, and how graphs behave relative to the x-axis.

The hardest problems were those that looked simple at first but hid subtle traps—like extraneous solutions or intervals that must be excluded. Solving them demanded patience, clear structure, and a solid understanding of why each step was valid.

By the end of the lesson, I realized that advanced inequalities are less about computation and more about logical control. Every inequality tells a story about how expressions behave, and solving them means understanding that behavior deeply. It was challenging, but extremely satisfying when everything finally fit together.