Problem 79 - Precalculus

Question

In Exercises \(79-84,\) determine whether each statement is true or false. A linear inequality always has a solution that is a half-plane.

Step-by-Step Solution

Answer

True, a linear inequality's solution is a half-plane.

Step 1: Understanding Linear Inequalities

A linear inequality in two variables, such as \(ax + by < c\), \(ax + by > c\), \(ax + by \leq c\), or \(ax + by \geq c\), defines a region of the coordinate plane. This region is called a half-plane, because it includes all the points on one side of the boundary line represented by the equation \(ax + by = c\).

Step 2: Visualizing the Graph

For a linear inequality, the boundary line is often dashed (for \(<\) and \(>\)) or solid (for \(\leq\) and \(\geq\)). The solution includes all the points on one side of this line. This side extends infinitely, creating a half-plane.

Step 3: Confirming the Half-Plane Concept

Since a linear inequality in two variables splits the plane into two regions, and the solution is all the points that satisfy the inequality in one of these regions, we can confirm that linear inequalities indeed have solutions that are half-planes.

Key Concepts

half-plane

A half-plane refers quite simply to one half of a coordinate plane, which is determined by a linear inequality. Picture the coordinate plane divided by a line; one side of this line, which continues infinitely in one direction, is the half-plane.
The line itself acts as a boundary, dividing the plane. Here are a few points to keep in mind regarding half-planes:

• A half-plane includes all points that satisfy the inequality.
• The line dividing it is not included in the half-plane if the inequality is strict (using \(<\) or \(>\)).
• For inequalities with \(\leq\) or \(\geq\), the line is included in the half-plane.

Understanding half-planes is crucial in analyzing solutions to linear inequalities, as it helps visualize how these solutions look on the coordinate plane.

boundary line

The boundary line is the line that separates the half-plane solution of a linear inequality from the rest of the coordinate plane. It's derived from the related linear equation \(ax + by = c\). This line is significant for a couple of reasons:

• If the inequality is \(<\) or \(>\), the boundary line is dashed, signifying it is not part of the solution.
• If the inequality is \(\leq\) or \(\geq\), the line is solid, indicating it is included in the solution.

To efficiently use the boundary line, one needs to decide on which side of the line a point should fall to satisfy the inequality. A common technique to determine this is by testing a point not on the line (often \((0,0)\) unless it lies on the line) to ascertain whether it fulfills the inequality.

coordinate plane

The coordinate plane is the foundational framework utilized in graphing linear inequalities. It is a two-dimensional space defined by a horizontal axis (the x-axis) and a vertical axis (the y-axis), intersecting at the origin \((0,0)\). The coordinate plane is essential for the following reasons:

• It houses the half-plane solutions to a linear inequality.
• It allows clear visualization of how linear inequalities divide space with a boundary line.
• It provides a canvas to plot and assess possible solutions by using the axes to measure distances and directions.

Understanding the coordinate plane helps in graphing both the boundary line and the half-plane, providing a spatial representation of all points satisfying the inequality. Knowing how to navigate the coordinate plane effectively is key to solving and comprehending linear inequalities.