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A feasible region is a core concept in systems of inequalities. It refers to the area on a graph where all the conditions of the inequalities are satisfied simultaneously.
When graphing a system of linear inequalities, each inequality will divide the plane into two halves - one where the inequality holds true, and one where it doesn't. The feasible region is the intersection of these true halves.
For example:

• If you have three inequalities, you will graph each one by determining if they use solid or dashed lines.
• Next, you'll find the area where all the shaded regions overlap.
• This overlapping region is what we call the feasible region.

This region may be bound by solid lines if the inequalities are non-strict (≥ or ≤), meaning the points on the lines are included. Or, it may be bounded by dashed lines if the inequalities are strict (< or >), indicating that points on these lines are not part of the feasible region.
In our example exercise, the feasible region is a triangular area where all the three given inequalities hold true. This region is important because it encompasses all possible solutions that satisfy the system of inequalities.

Inequality graphing involves plotting each inequality on a coordinate plane to visualize the solution set.
The process starts by transforming each inequality into an equation (e.g., replacing ≥ with =), allowing you to graph a boundary line.
For example:

• Equation of the form \( y \geq x - 1 \) becomes the line \( y = x - 1 \).
• Once plotted, determine if the line should be solid or dashed. Solid lines are used if the inequality is non-strict (≥ or ≤), indicating that points on the line are included in the solution. Dashed lines represent strict inequalities (< or >), where points on the line are excluded.

After graphing the line, you determine on which side of the line the inequality holds true and shade that region. For instance, if the inequality is \( y \geq x - 1 \), you would shade above the line since that represents all points where \( y \) values are greater than or equal to \( x - 1 \).
This method helps in visualizing the potential solutions for each inequality in the system.

Shading regions is a vital step in graphing systems of inequalities as it displays where each inequality holds true.
Once you plot the boundary line for each inequality, decide which region or side of the line to shade. This region contains all the possible solutions for that inequality.
Steps to shade regions accurately:

• First, choose a test point that is not on the line to determine which side should be shaded. Common choices are points like (0, 0) unless it lies on the line.
• If the test point satisfies the inequality, shade the side where it lies. If not, shade the opposite side.

In the example exercise given:

• For the inequality \( y \geq x - 1 \), shade the region above the line since points there have greater y-values.
• For \( y \leq -x + 3 \), shade below the line as y-values in this region are smaller.
• Lastly, for \( y < x + 2 \), a dashed line is used. Here, shade the area below this line, indicating that the actual line is not included in the solution.

Thus, shading helps you easily identify where conditions from all inequalities overlap, leading to the feasible region.