By Ariel Balter, Ph.D. Updated Aug 30, 2022
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Linear programming is a cornerstone of modern business analytics, enabling companies to optimize resources by solving systems of inequalities. In this guide, we’ll walk through the core techniques for shading inequalities—both in one‑dimensional number‑line problems and two‑dimensional coordinate‑plane problems—so you can apply these skills confidently in real‑world contexts.
Consider the inequality x < 5. The symbol “<” excludes the value 5, so we mark an open circle at 5. The number line splits into two regions: values less than 5 (to the left) and values greater than 5 (to the right). Test a point from the left region, such as 0. Since 0 < 5, we shade the entire left side, extending beyond 0 to the left.
Now add the constraint x >= -3. Because “≥” includes –3, we draw a solid circle at –3. Testing a point in the overlapping region—say 0—confirms 0 ≥ –3, so we shade the region that contains 0, which lies to the right of –3 but still left of the open circle at 5. The final shaded area is the intersection of the two conditions.
In two dimensions, we use solid and dashed lines to represent boundary conditions. For x = 5 (strict inequality), draw a dashed vertical line. For x = -3 (inclusive), draw a solid vertical line. Shade the strip between them. For an inequality such as y < -2x + 3, graph the line y = -2x + 3 with a dashed line. Pick a test point on one side of the line—e.g., (3, 4). Substituting gives 4 < 9, which is true, so shade the side containing (3, 4). If the test failed, shade the opposite side.
