Problem 51 - Introductory Algebra for College Students

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 statement about a region of the coordinate plane comprising all solutions to the inequality. It can be written as \( ax + by > c \), \( ax + by < c \), \( ax + by \geq c \), or \( ax + by \leq c \). An example is \(2x + 3y > 6 \)

Step 1: Define Linear Inequality

A linear inequality in two variables is a statement that represents a region of the coordinate plane that includes all solutions to the inequality. The simplest form of it can be written as \( ax + by > c \), \( ax + by < c \), \( ax + by \geq c \), or \( ax + by \leq c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. The inequality symbol can be , < , > , ≤ , or ≥.

Step 2: Provide an Example

An example of a linear inequality in two variables is \(2x + 3y > 6 \). The solution set for this inequality is a half-plane made of points that satisfy the inequality. Hence any solution can be a point \( (x, y) \) in this half-plane.