Question

Explain what a solution of a linear inequality in \(x\) and \(y\) is.

Short answer

A solution of a linear inequality in \(x\) and \(y\) is the ordered pair that makes the inequality true. Mathematically, the solution set of an inequality in two variables \(x\) and \(y\) is often represented graphically as a region on the coordinate plane.

Step-by-step solution

Definition of a Linear Inequality

In mathematics, a linear inequality is an inequality which involves a linear function. A linear function is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. In this case, the inequality is in two variables, \(x\) and \(y\). It will have a general form of \(ax + by > c\) or \(ax + by < c\) or \(ax + by ≥ c\) or \(ax + by ≤ c\), where a, b, and c are constants.

What is a Solution of a Linear Inequality

The solution of a linear inequality is the ordered pair (that is, the pair of numbers), that makes the inequality true when the values of \(x\) and \(y\) in the ordered pair are substituted into the inequality. Generally it is represented graphically as a region of the coordinate plane that is bounded by the line corresponding to the equality of the given inequality.

Graphical Representation of Solution

The solution set to a linear inequality is graphed on a coordinate plane. The line represented by the associated equation (with '=') will divide the plane into two halves. Depending on whether the inequality is 'less than' or 'greater than', and considering if it's 'equal to', one of these halves is shaded to represent all possible solutions. If the inequality is 'less than or equal to' or 'greater than or equal to', the line itself is also part of the solution, hence it's drawn with a solid line. If it's strictly 'less than' or 'greater than', the line is not part of the solution, and it's drawn as a dashed line.