Regions defined by inequalities

Japanese school year: Math II

What you learn

This section teaches how to graph regions represented by inequalities on the coordinate plane and solve optimization problems like linear programming. These graphical methods are widely used to identify feasible regions under constraints. Prior familiarity with linear equations, circle equations, and basic inequality properties is recommended.

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Key points

This inequality represents the region lying above the boundary curve $y=f(x)$ . For any given $x$ , it includes all points with a larger $y$ -coordinate; reversing the inequality to $y < f(x)$ gives the region below.

y > f(x)

This inequality represents the open interior region of a circle with center $(a, b)$ and radius $r$ . It collects points whose distance to the center is less than $r$ , whereas $> r^2$ defines the exterior.

(x - a)^2 + (y - b)^2 < r^2

This rule clarifies whether the boundary is part of the plotted region. If the inequality includes an equals sign ( $\le$ or $\ge$ ), state that the boundary is included; for strict inequalities ( $<$ or $>$ ), state that the boundary is excluded.

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