Two- Variable Inequalities

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

To graph a two-variable inequality, first graph the corresponding equation as a line. Then, use a dashed line for < or > and a solid line for ≤ or ≥. Finally, shade the region that satisfies the inequality.

The shaded region represents all the possible solutions (x, y) that satisfy the inequality.

A solid line indicates that points on the line are included in the solution (≤ or ≥), while a dashed line indicates that points on the line are not included ( < or > ).

To determine which side to shade, pick a test point not on the line (commonly (0,0)). Substitute it into the inequality. If the inequality holds true, shade the side containing the test point; otherwise, shade the opposite side.

The standard form of a linear inequality is written as $$Ax + By < C$$ , $$Ax + By > C$$ , $$Ax + By \leq C$$ , or $$Ax + By \geq C$$ , where A, B, and C are constants.

The inequality $$y \geq mx + b$$ represents all points (x, y) that lie on or above the line defined by the equation $$y = mx + b$$ .

The slope-intercept form of a linear equation is $$y = mx + b$$ , where m is the slope and b is the y-intercept.

To convert a linear equation to an inequality, replace the equality sign (=) with an inequality sign (<, >, ≤, or ≥) based on the context of the problem.

The y-intercept is the point where the line crosses the y-axis, and it helps to determine the starting point for graphing the inequality.