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 the inequality $a x+b y+c \leq 0$ is either a left half-plane or a lower half-plane.

Key Concepts

Linear Inequalities in Two Variables

A linear inequality in two variables, such as ax + by + c ? 0, defines a set of points in the coordinate plane that satisfy the inequality. The boundary of this set is a straight line given by ax + by + c = 0, and the inequality determines which side of the line is included in the solution set.

Half-Planes

The solution set of a linear inequality in two variables is always one of the two half-planes divided by its boundary line. A half-plane is the region on one side of a line that extends infinitely in all directions within that side. Importantly, the specific half-plane is not restricted to being only vertically (left/right) or horizontally (upper/lower) oriented; it can be inclined at any angle depending on the coefficients a and b.

Boundary Lines

The boundary line, defined by the equation ax + by + c = 0, plays a crucial role in determining the solution set of the inequality. It separates the plane into two regions (half-planes) and is included in the solution set when the inequality is non-strict (using ? or ?). Understanding the orientation and position of this line is essential to analyzing the nature of the half-plane solution.

Use of Counterexamples in Mathematical Reasoning

In mathematics, counterexamples are used to disprove general statements by providing a specific case where the statement fails. In this context, to show that the statement about the solution set being only a left or lower half-plane is false, one might present an inequality whose boundary line is not vertical or horizontal, such as x + y - 1 ? 0, illustrating that the half-plane can have any orientation.

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