📊 Linear Inequalities: 🎓 Final Exam

An open circle (○) indicates a strict inequality ($<$ or $>$) where the boundary point is not included. A solid circle (●) indicates an inclusive inequality ($\le$ or $\ge$) where the boundary point is part of the solution.

When the variable is isolated on the left side (like $x > a$ or $x \ge a$), shade to the right towards positive infinity ($+\infty$). For $x < a$ or $x \le a$, shade to the left towards negative infinity ($-\infty$). You can check the direction by choosing a test point in the region you expect to be the solution, such as $x = 0$ when appropriate (which indicates your shading is probably correct, though testing one point is not definitive proof).

"At least" represents a minimum requirement and translates to greater than or equal to ($\ge$). "At most" represents a maximum limit and translates to less than or equal to ($\le$).

You flip the inequality sign (e.g., from $<$ to $>$, or from $\le$ to $\ge$) whenever you multiply or divide both sides by a negative number. Adding or subtracting numbers, or multiplying/dividing by a positive number, never flips the sign.

A compound inequality $a < x \le b$ represents the interval where $x$ is strictly greater than $a$ and at most $b$. To graph it, place an open circle at $a$, a solid circle at $b$, and shade the line segment between them.

Pick a test point from within your shaded solution region (such as $x = 0$ or $x = 10$) and substitute it into the original problem. If it yields a true statement (such as $5 \le 12$), your solution and shading are probably correct (though a single test point is not a definitive proof).