Two- Variable Inequalities

A two-variable inequality is an inequality that involves two variables, typically written in the form of $$ax + by < c$$ , $$ax + by > c$$ , $$ax + by \leq c$$ , or $$ax + by \geq c$$ .

The graph of a two-variable inequality represents all the solutions (ordered pairs) that satisfy the inequality, often shown as a shaded region on a coordinate plane.

The boundary line is the line that represents the equation formed by replacing the inequality sign with an equal sign. It divides the plane into two regions, one of which contains the solutions to the inequality.

The boundary line is included in the solution set if the inequality is either $$\leq$$ or $$\geq$$ .

The boundary line is not included in the solution set if the inequality is either < or >.

To determine if a point (x, y) is a solution, substitute the x and y values into the inequality. If the inequality holds true, then the point is a solution.

It represents a region above the line $$y = 2/5x + 1$$ , including the line itself.

It represents a region below the line $$y = -3/4x + 3$$ , including the line itself.

The solution set includes all points above the parabola defined by the equation, not including the points on the parabola itself.

Strict inequalities (<, >) do not include the boundary line, while non-strict inequalities (\leq, \geq) do include the boundary line.