How can you identify the solution to a system of inequalities?

The solution to a system of inequalities is identified graphically as the region where the solutions to all inequalities in the system overlap.

Here's a step-by-step breakdown of how to identify the solution:

1. Graph Each Inequality:

   • Treat each inequality as if it were an equation (replace the inequality sign with an equals sign) and graph the corresponding line. This line is the boundary line.
   • If the inequality is strict (using < or >), the boundary line is dashed to indicate that points on the line are not included in the solution.
   • If the inequality includes equality (using ≤ or ≥), the boundary line is solid to indicate that points on the line are included in the solution.

2. Shade the Solution Region for Each Inequality:

   • For each inequality, choose a test point not on the boundary line (the origin (0,0) is often easiest).
   • Substitute the test point's coordinates into the original inequality.
   • If the inequality is true, shade the region containing the test point. This region represents all points that satisfy the inequality.
   • If the inequality is false, shade the region not containing the test point.

3. Identify the Overlapping Region:

   • The solution to the system of inequalities is the region where the shaded areas of all the inequalities overlap. This overlapping region contains all the points that satisfy all inequalities in the system simultaneously.
   • If there is no overlapping region, the system has no solution.

Example:

Consider the system of inequalities:

• y > x + 1
• y ≤ -x + 4

1. Graphing: You'd graph the lines y = x + 1 (dashed) and y = -x + 4 (solid).
2. Shading: For y > x + 1, testing (0,0) gives 0 > 0 + 1, which is false. So, shade the region above the dashed line. For y ≤ -x + 4, testing (0,0) gives 0 ≤ -0 + 4, which is true. So, shade the region below the solid line.
3. Overlapping Region: The solution is the region where the two shaded regions overlap. This region represents all (x, y) pairs that satisfy both inequalities.

In summary, the graphical solution to a system of inequalities is the intersection of the solution regions of each individual inequality. You find this by graphing each inequality and identifying the region where all shaded areas overlap.