Graphing Systems of Inequalities and Solution Regions

A linear inequality in two variables describes a half plane. Its boundary is the corresponding line.

ax + by \le c,\quad ax + by < c,\quad ax + by \ge c,\quad ax + by > c.

Boundary style:

Use a solid line for $\le$ or $\ge$ because points on the line satisfy the inequality.
Use a dashed line for $<$ or $>$ because boundary points are not included.

Shading rule:

Convert to $y = mx + b$ if convenient. For $y \ge mx + b$ , shade above the line. For $y \le mx + b$ , shade below.
Or use a test point that is not on the boundary, often $(0,0)$ . If it makes the inequality true, shade the side containing the test point.

A system of inequalities is satisfied by points that make every inequality true. The solution region is the intersection of the shaded half planes.

Core Skills

Rearrange each inequality to identify the boundary and inequality direction.
Draw the boundary with correct style, then use a test point to decide shading.
Find the intersection region. State whether it is bounded or unbounded and list vertices if needed.

Example 1: Single Inequality

Graph

y \le -2x + 5

. Boundary:

y = -2x + 5

with solid line since

\le

. Shading: for

y \le

shade below the line. Check with

(0,0)

0 \le 5

is true, so the origin side is shaded.

Example 2: Two Inequalities

Graph the system

\begin{cases} y > x - 2\\ y \le -\tfrac{1}{2}x + 4 \end{cases}

First boundary:

y = x - 2

dashed, shade above. Second boundary:

y = -\tfrac{1}{2}x + 4

solid, shade below. The solution region is where the two shadings overlap. Find intersection by solving the equalities:

x - 2 = -\tfrac{1}{2}x + 4 \Rightarrow \tfrac{3}{2}x = 6 \Rightarrow x = 4,\quad y = 2.

The corner point is

(4,2)

, the intersection of both boundaries. Since the first boundary is strict

>

(4,2)

is not included in the solution set.

Example 3: With a Vertical or Horizontal Boundary

Graph

\begin{cases} x \ge 1\\ y < 3 \end{cases}

Boundary

x=1

is a vertical solid line. Shade to the right. Boundary

y=3

is a horizontal dashed line. Shade below. The solution region is the unbounded rectangle corner with vertex

(1,3)

but that point is excluded because

y<3

Key Takeaways

Solid for inclusive $\le,\ \ge$ , dashed for strict $<,\ >$ .
Use slope intercept form or a test point to choose the correct side.
The solution to a system is the intersection of shadings. State whether boundary points are included.
Vertical lines use $x=\text{constant}$ . Horizontal lines use $y=\text{constant}$ .