Graph linear inequalities in two variables

Detailed Explanation: Graph linear inequalities in two variables

A linear inequality in two variables is graphed by showing all points $(x,y)$ that make the inequality true. The graph has two parts:

1. A boundary line.
2. A shaded half-plane on one side of the line.

Use this example:

Step 1: Graph the boundary line

Replace the inequality symbol $>$ with an equals sign:

Solve for $y$:

This line has a $y$-intercept of $4$ and a slope of $-2$. Plot two points, such as:

• $(0,4)$
• $(2,0)$

Because the original inequality uses $>$, equality is not included. Therefore, draw the boundary line dashed.

Step 2: Choose a test point

Choose a point that is not on the boundary line. The origin, $(0,0)$, is convenient.

Substitute $x=0$ and $y=0$ into the original inequality:

This is false, so $(0,0)$ is not part of the solution region.

Step 3: Shade the correct side

Since the origin is on the side that does not work, shade the opposite side of the dashed line.

Equivalently, rewrite the inequality as:

This means shade the region above the dashed line.

Every point in the shaded region makes $2x+y>4$ true. Points on the dashed line are not included because the inequality is strict. If the inequality had been $\geq$ or $\leq$, the boundary would be drawn solid.