91Ó°ÊÓ

In mathematics, an inequality is a relationship between two expressions. These expressions can either be constants, variables, or a combination of both. Inequalities tell us about the relative size of two values. Here are the symbols used:

• \(<\): less than
• \(\leq\): less than or equal to
• \(>\): greater than
• \(\geq\): greater than or equal to

In a system of inequalities, we have to consider multiple inequalities simultaneously. Each inequality can define a region on a graph, and we aim to find where all these regions overlap. This overlapping region is called the solution set.

Graphing inequalities involves visualizing the solutions on a coordinate plane. Each inequality is represented by a line or curve, along with shading to show the feasible region. Let's break it down with an example:
First, consider the inequality \(-2 < x < 3\). This can be visualized by drawing vertical dashed lines at \(x = -2\) and \(x = 3\). The dashed lines indicate that the points on these lines are not included in the solution. The area between these lines must be shaded to represent the solutions for \(x\).

Next, for the inequality \(-1 \leq y \leq 5\), draw horizontal solid lines at \(y = -1\) and \(y = 5\). Solid lines indicate that the boundary points are included. Shade the region between these lines.

Finally, the inequality \(2x + y < 6\) needs to be graphed. Rewrite it as \(y < -2x + 6\), which makes it easier to graph. Draw a dashed line for \(y = -2x + 6\), and shade the area below this line. The shading below indicates that solutions fall under this line.

The solution set for a system of inequalities is the region where all shaded areas overlap. This overlap represents all pairs of \((x, y)\) that satisfy every inequality in the system. For our example:
1. The vertical dashed lines at \(x = -2\) and \(3\) define a narrow vertical strip.
2. The horizontal solid lines at \(y = -1\) and \(y = 5\) depict a wide horizontal strip.
3. The dashed line from \(y = -2x + 6\) with shading below it constrains the region even further.
The common area where these three regions intersect is the solution set. This region contains all the points \((x, y)\) that make each inequality true simultaneously. Identifying this region visually helps significantly in understanding the solutions for a system of inequalities.