Problem 87 - College Algebra

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 an inequality that can be expressed in the form Ax + By > C, Ax + By < C, Ax + By ≥ C, or Ax + By ≤ C, where A, B, and C are real numbers and x and y are variables. The solutions to such inequalities form regions on the coordinate plane. An example is the inequality 2x + 3y ≤ 6.

Step 1: Define linear inequality

A linear inequality in two variables is an inequality that can be written in the form Ax + By > C, Ax + By < C, Ax + By ≥ C, or Ax + By ≤ C. Here, A, B, and C are real numbers and x and y are variables.

Step 2: Explain nature of the inequality

The solution to such an inequality is not just a single point, but a region of solutions. It means that any point (x, y) that makes the inequality true is part of the solution. These regions are graphically represented on a two-dimensional coordinate plane.

Step 3: Provide an example

Assume the inequality 2x + 3y ≤ 6. It represents an infinite number of ordered pairs (x, y) that satisfy the inequality, collectively forming a region on the coordinate plane. In the case of this example, it would represent all the points below or on the line 2x + 3y = 6. We can verify the solution by picking a point from the region. For example, the point (1,1) since 2(1) + 3(1) = 5 which is indeed less than or equal to 6.