Determine whether the statement is true or false. If it is...

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
The solution set of a linear inequality involving two variables is either a half-plane or a straight line.

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Key Concepts

Degenerate Cases

Sometimes a linear inequality can degenerate when the coefficients of the variables are zero or lead to an identity or contradiction. For example, if the inequality reduces to a statement about constants (such as 0 ? c), the solution set might be the entire plane or the empty set, neither of which is a half-plane or a straight line. These cases show that the general statement does not always hold true.

Linear Inequality

A linear inequality in two variables is an inequality that contains a linear expression (such as ax + by) compared to a constant using inequality symbols (like <, ?, >, or ?). Typically, for non-degenerate coefficients, it represents a constraint whose boundary is given by the corresponding linear equation, dividing the plane into two regions.

Half-Plane

A half-plane is one of the two regions created when a straight line divides the coordinate plane. In the context of a linear inequality (when the coefficients of the variables are not all zero), the solution set is usually the half-plane defined by the linear boundary, possibly including the boundary line if the inequality is non-strict (i.e., ? or ?).

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