Graphing Linear Inequalities and Their Intersections in Two Variables

Graphs and Functions

Graphing Linear Inequalities

Linear inequalities in two variables are foundational in algebra and are used to describe regions of the coordinate plane that satisfy certain conditions. These inequalities are typically written in the form $Ax + By < C$, where $A$, $B$, and $C$ are real numbers and $A$ and $B$ are not both zero. The solution to a linear inequality is a set of ordered pairs $(x, y)$ that make the inequality true.

• Definition: A linear inequality in two variables is an inequality that can be written as $Ax + By < C$, $Ax + By \leq C$, $Ax + By > C$, or $Ax + By \geq C$.

• Solution: An ordered pair $(x, y)$ is a solution if substituting $x$ and $y$ into the inequality yields a true statement.

Steps for Graphing a Linear Inequality

1. Graph the Boundary Line: Replace the inequality sign with an equal sign to obtain the boundary line. If the inequality is $<$ or $>$, use a dashed line (points on the line are not included). If the inequality is $\leq$ or $\geq$, use a solid line (points on the line are included).

2. Choose a Test Point: Select a point not on the boundary line (often $(0,0)$ if possible) and substitute its coordinates into the original inequality.

3. Shade the Appropriate Region: If the test point satisfies the inequality, shade the half-plane containing the test point. Otherwise, shade the opposite half-plane.

Example 1: Graph $x - y > 3$

• Step 1: Graph the boundary line $x - y = 3$ using a dashed line.

• Step 2: Test the point $(0,0)$: $0 - 0 > 3$ is false.

• Step 3: Shade the half-plane that does not contain $(0,0)$.

Example 2: Graph with a Solid Boundary Line

• Step 1: Use a solid boundary line (for $\leq$ or $\geq$ inequalities).

• Step 2: Use $(0,0)$ as a test point. If true, shade the side containing $(0,0)$.

Example 3: Graph with a Solid Boundary Line and False Test Point

• Step 1: Graph the solid boundary line.

• Step 2: Test $(0,0)$; if false, shade the half-plane that does not contain $(0,0)$.

Helpful Hint

• Always substitute the test point into the original inequality.

• If the boundary line passes through $(0,0)$, choose a different test point not on the line.

Graphing Intersections or Unions of Two Linear Inequalities

When working with two linear inequalities, the solution set can be found by graphing both inequalities and identifying either the intersection (overlap) or the union (combined shaded regions) of their solution sets.

• Intersection: The set of points that satisfy both inequalities. This region is where the shaded areas of both inequalities overlap.

• Union: The set of points that satisfy at least one of the inequalities. This region includes all points shaded by either inequality.

Example 7: Intersection of Two Inequalities

• Graph each inequality separately.

• The intersection is the region common to both shaded areas (often shown as a different color or shade).

Example 8: Union of Two Inequalities

• Graph each inequality separately.

• The union is the combined area of both shaded regions.

Summary Table: Boundary Line Types

Inequality Symbol	Boundary Line Type	Points on Line Included?
< or >	Dashed	No
≤ or ≥	Solid	Yes

Additional info: Mastery of graphing linear inequalities is essential for solving systems of inequalities and for applications in optimization, such as linear programming.