Problem 93 - Algebra and Trigonometry

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 inequalities with the same set of variables. The solution to this system is the area where all the inequalities overlap when graphed.

Step 1: Understand Linear Inequality

A linear inequality is similar to a linear equation, but it uses inequality symbols rather than equality. For example, \(2x + 5y > 10\) is a linear inequality. This inequality includes all values of x and y for which this statement holds true.

Step 2: Graphical Representation

Each linear inequality represents a region on the graph, separated by a line. The line is solid if the inequality includes equality (like \(\leq\) or \(\geq\)), and it is dashed if not (\(>\) or \(<\)). The side of the line that contains solutions is determined by picking a test point (often the origin, if it's not on the line) and checking whether the inequality holds.

Step 3: System of Linear Inequalities

A system of linear inequalities will have more than one inequality. For example, a system could be: \(2x + 5y > 10\) and \(3x - y < 2\). When dealing with a system, it is required to find the solution to both inequalities, meaning finding the overlapping region in the graph.

Step 4: Solve the System

The solution to the system is the set of points (x, y) that satisfy all the inequalities in the system. To find the solution, each inequality is graphed, and the region where all the solutions overlap is the solution to the system.