#### Chapter 9 Objects in the Two-Dimensional Coordinate System

**Section 9.4 Regions in the plane**

# 9.4.1 Introduction

While in the previous sections curves in the plane (lines or circles) were investigated by means of coordinate equations, in this section we will replace the coordinate

*equations*by coordinate

*inequalities*. Thereby not curves but

*regions*in the plane are described, which are bounded by the corresponding curves. Depending on whether the inequality is strict ($<$ or $>$) or not ($\le $ or $\ge $), the bounding curve is a part of the region or not. Regions can be, for example, areas above and below lines, areas within or outside circles, or even intersections of these. A few examples are shown in the figures below.

- Region above the line $y=\frac{1}{2}x-1$ excluding the line itself:

- Region below the line $y=2$ including the line itself:

- Region above the line $y=x$ and within the unit circle ${x}^{2}+{y}^{2}=1$ including the points on the circle but excluding the points on the line:

*excluded*from the region are drawn as dashed lines.