How do you solve systems of inequalities in two variables?
To solve systems of inequalities in two variables, follow these steps:
Step 1: Graph each inequality separately on the coordinate plane. Treat each inequality as if it were an equation, and graph the associated line or boundary. Use a solid line for "≤" or "≥" inequalities, and a dashed line for "<" or ">" inequalities.
Step 2: Determine the shaded area for each inequality. Select a test point not on the line and substitute its coordinates into the inequality. If the inequality is satisfied, shade the region containing the test point; otherwise, shade the other region. Repeat this process for all inequalities.
Step 3: Identify the overlapping shaded area. The solution to the system of inequalities is the region where all the shaded areas overlap.
Step 4: Write the solution as an ordered pair or a range of values. You can represent the solution as (x, y) coordinates or inequalities stating the range of values for the variables.
It's important to note that the solution to a system of inequalities may not always be a single point but could be an infinite set of solutions or even an empty set, depending on the specific problem.