Problem 87 - Algebra and Trigonometry

Question

What is a linear inequality in two variables? Provide an example with your description.

Step-by-Step Solution

Answer

A linear inequality in two variables is a relationship between two variables that can be graphically represented as a line, along with a shaded region. An example would be the inequality \(y > 2x + 3\), where the solution region is the area above the line \(y = 2x + 3\)

Step 1: Definition of Linear Inequality in Two Variables

A linear inequality in two variables is a relationship between two different variables that forms a region in the coordinate plane. This inequality can be represented on a graph as a line and a shaded region either above or below the line. The line divides the plane into two halves, and the inequality specifies which half-plain is considered the solution region. A point in this region will satisfy the inequality, whereas a point outside of it will not.

Step 2: Example of Linear Inequality in Two Variables

For example, consider the inequality \(y > 2x + 3\). This is a linear inequality in two variables. To graph this, begin by drawing the line \(y = 2x + 3\). Since the inequality symbol is '>', this means that the solution region is above the line. So, the region above the line (excluding the line itself, since it's not \(y \geq 2x + 3\)) is shaded to indicate the solution region. Any point within this shaded region fulfills the inequality.