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 ≤ c', or 'ax + by ≥ c'.

The solution set of a two-variable inequality represents all the ordered pairs (x, y) that satisfy the inequality, often visualized as a shaded region on a graph.

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

The boundary line represents the points where the inequality becomes an equation. If the inequality is strict (< or >), the line is dashed; if it is inclusive (≤ or ≥), the line is solid.

The graph of 'y < -4x - 1' is a dashed line with a slope of -4 and a y-intercept of -1, with the area below the line shaded.

To graph '4y > -8', first rewrite it as 'y > -2'. Then, draw a dashed horizontal line at y = -2 and shade the area above the line.

'4y < -2' represents the area below the line y = -0.5, while '4y > -2' represents the area above the same line.

The inequality '4y - 3x > 20' represents a region above the line 4y = 3x + 20, which is a solid line if the inequality is inclusive.

Test points are used to determine which side of the boundary line to shade. By substituting a test point into the inequality, you can see if the point satisfies the inequality.

A dashed line indicates that the points on the line are not included in the solution set for strict inequalities (< or >).