일차함수와 일차방정식

The line $y=2x-1$ is the boundary line. On one side of the line are the points with $y>2x-1$ and on the other side of the line are the points with $y<2x-1$.

Let’s test the point $(0,0)$ and see which inequality describes its side of the boundary line.

At $(0,0)$, which inequality is true:

$$\begin{array}{ccccc}y>2x-1 & & \text{or} & & y<2x-1? \\ y>2x-1 & & & & y<2x-1 \\ 0\overset{?}{>}2\cdot 0-1 & & & & 0\overset{?}{<}2\cdot 0-1 \\ 0>-1\,\text{True} & & & & 0<-1\,\text{False}\end{array}$$

Since, $y>2x-1$ is true, the side of the line with $(0,0)$, is the solution. The shaded region shows the solution of the inequality $y>2x-1$.

Since the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality $y\ge 2x-1$.

We could use any point as a test point, provided it is not on the line. Why did we choose $(0,0)$? Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that $y<2x-1$.