1. Advanced Inequality Mindset

Advanced inequalities are not about memorizing more symbols. They are about tracking regions of possible answers.

An equation usually asks, "Which value makes this exactly true?"

An inequality asks, "Which values make this comparison true?"

That means the final answer is often an interval, a union of intervals, or a shaded region.

Core habits:

• Keep the inequality balanced.
• Flip the sign only when multiplying or dividing by a negative.
• Watch whether endpoints are included.
• Test values when signs can change.
• Check the answer against the original inequality.

Example:
\[ (x-2)(x+3)>0. \]
The expression is zero at \(x=2\) and \(x=-3\). These values split the number line into regions:

• \(x<-3\)
• \(-3
• \(x>2\)

Test one value in each region. The product is positive on the outside regions, so:
\[ x<-3 \quad \text{or} \quad x>2. \]

Common mistake: solving only the zero points and forgetting to decide where the inequality is true.