Inequality regions

Objective. Model feasible regions from several linear constraints.

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Learn: the key idea

Inequality regions model real situations by drawing several linear inequalities on one set of axes and shading the area satisfying all of them simultaneously. Draw each boundary line (solid for ≤ or ≥, dashed for < or >), decide which side satisfies the inequality by testing a point such as the origin, then identify the overlap of all shaded sides as the feasible region. For example, y ≤ x + 2 and x + y ≥ 1 together define a wedge-shaped region bounded by both lines.

Worked example

Describe the feasible region satisfying the inequality regions x ≥ 0, y ≥ 0 and x + y ≤ 6, then state whether the point (2, 3) lies inside it.

1 $Boundaries are the axes x = 0, y = 0 and the line x + y = 6$
2 $Region requires x \geq 0, y \geq 0 (first quadrant) and below the line x + y \leq 6$
3 $Test (2,3): both coordinates are non-negative, satisfying the axis conditions$
4 $Check x + y = 2 + 3 = 5 \leq 6, so the inequality also holds$

Answer: (2, 3) lies inside the feasible region

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Common mistake. A frequent mistake is shading the wrong side of a boundary line by guessing instead of testing a coordinate pair in the original inequality. Always substitute a simple test point like (0,0) into each inequality to confirm which side is correct.

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