Problem 93 - College Algebra

Question

What is a system of linear inequalities?

Step-by-Step Solution

Answer

A system of linear inequalities is a set of two or more linear inequalities having the same variables. The solution is the intersection of the solutions to each inequality, which usually forms a geometric shape.

Step 1: Understand Linear Inequality

A linear inequality is like a linear equation but instead of an equals sign, it has an inequality sign. For instance, \(2x + 3y > 6\) is a linear inequality. The solution to a linear inequality is a range of values, not just a singular point.

Step 2: Understanding Half-Spaces

A linear inequality divides the coordinate plane into two halves, called half-spaces. For example, in the inequality \(2x + 3y > 6\), the solutions lie in the half-space where \(2x + 3y\) is more than 6. When graphed, this half-space does not include the line \(2x + 3y = 6\). If the inequality was \(2x + 3y \geq 6\), the line would be included in the half-space, and it would consist of all points that make the inequality true when substituted.

Step 3: Understand a System of Linear Inequalities

A system of linear inequalities is a set of two or more linear inequalities with the same variables. The solution to this system is the intersection of the solutions to each inequality. This usually forms a geometric shape, graphically represented by the area where the half-spaces intersect.