What Are Linear Inequalities?
A linear inequality is similar to a linear equation but uses inequality symbols (<, >, ≤, ≥) instead of equals. The solution to a linear inequality is a region of the coordinate plane, not just a line.
y > mx + b → Shade above the dashed line
y < mx + b → Shade below the dashed line
y ≥ mx + b → Shade above the solid line
y ≤ mx + b → Shade below the solid line
How to Graph Inequalities
- Graph the boundary line (y = mx + b)
- Use a dashed line for < or > (boundary not included)
- Use a solid line for ≤ or ≥ (boundary included)
- Test a point (usually origin) to determine which side to shade
- Shade the region containing all solutions
Systems of Inequalities
When graphing a system (multiple inequalities), the solution is the region where all shaded areas overlap. This region satisfies all inequalities simultaneously.
How to Use
Enter the slope (m), y-intercept (b), and inequality type. The calculator determines whether points satisfy the inequality.
Applications
Linear inequalities model real-world constraints: budget limits, production capacities, nutritional requirements, and resource allocation. They form the foundation of linear programming.
Test Point Method
To determine which side of the line to shade, substitute a test point (0,0 if it is not on the line) into the inequality. If it satisfies the inequality, shade that side; otherwise, shade the opposite side.