Two-Variable Inequalities

A two-variable inequality is a mathematical statement that compares two expressions involving two variables, typically written in the form $$ax + by < c$$ , $$ax + by > c$$ , $$ax + by \leq c$$ , or $$ax + by \geq c$$ .

Graphing a two-variable inequality involves representing the solutions of the inequality on a coordinate plane, typically using a boundary line and shading the appropriate region.

The boundary line represents the points where the inequality becomes an equation. If the inequality is strict (e.g., < or >), the line is dashed; if it is inclusive (e.g., \leq or \geq), the line is solid.

A dashed line is used when the inequality is strict, meaning the points on the line are not included in the solution set (e.g., $$y < mx + b$$ or $$y > mx + b$$ ).

A solid line is used when the inequality is inclusive, meaning the points on the line are included in the solution set (e.g., $$y \leq mx + b$$ or $$y \geq mx + b$$ ).

Shading above the line indicates that the solutions to the inequality are greater than the boundary line (e.g., for $$y > mx + b$$ or $$y \geq mx + b$$ ).

Shading below the line indicates that the solutions to the inequality are less than the boundary line (e.g., for $$y < mx + b$$ or $$y \leq mx + b$$ ).

To determine if a point is a solution, substitute the x and y values of the point into the inequality. If the inequality holds true, the point is part of the solution set.

The solution set of a two-variable inequality is the set of all points (x, y) that satisfy the inequality, represented graphically by the shaded region on the graph.

A strict inequality (e.g., < or >) does not include the boundary line, while a non-strict inequality (e.g., \leq or \geq) includes the boundary line.