Linear inequalities in two variables divide the coordinate plane into regions, with one region representing the solution set. Graphing these inequalities involves drawing boundary lines and shading the appropriate half-plane based on the inequality symbol and a test point.

Understanding linear inequalities is crucial for solving real-world problems involving constraints. This knowledge forms the foundation for linear programming, where multiple inequalities create a feasible region, allowing us to optimize solutions within given limitations.

Graphing Linear Inequalities in Two Variables

Verification of two-variable inequality solutions

• Substitute the given values for the variables into the inequality
  • Inequality true after substitution point is a solution
  • Inequality false after substitution point is not a solution
• Given inequality $3x + 2y < 12$, verify if (2, 3) is a solution
  • Substitute $x = 2$ and $y = 3$: $3(2) + 2(3) < 12$
  • Simplify: $6 + 6 < 12$ or $12 < 12$, which is false
  • (2, 3) is not a solution to the inequality

Interpretation of linear inequality graphs

• Linear inequality graph divides coordinate plane into two regions (half-planes)
  • One region represents points satisfying inequality (solution set)
  • Other region represents points not satisfying inequality
• Boundary line determined by corresponding linear equation
  • Inequality symbol $<$ or $>$ boundary line dashed (not included in solution set)
  • Inequality symbol $\leq$ or $\geq$ boundary line solid (included in solution set)
• Shaded region represents solution set of inequality
  • Test point not on boundary line to determine shading
  • Test point satisfies inequality shade region containing it
  • Test point does not satisfy inequality shade region not containing it

Graphing of linear inequalities

• Convert inequality to slope-intercept form: $y < mx + b$ or $y > mx + b$
• Graph corresponding boundary line $y = mx + b$
  • Dashed line for strict inequalities ($<$ or $>$)
  • Solid line for inclusive inequalities ($\leq$ or $\geq$)
• Choose test point not on boundary line (0, 0)
  • Substitute test point into inequality
  • Inequality true shade region containing test point
  • Inequality false shade region not containing test point
• Label graph with the inequality

Real-world applications of linear inequalities

• Identify variables and meanings in problem context
• Write inequality representing constraints or conditions in problem
  • Pay attention to direction of inequality symbol
• Graph inequality on coordinate plane
  • Interpret solution set in problem context
• Company produces two products A and B
  • Product A requires 2 hours, product B requires 3 hours
  • Maximum 24 hours available for production
  • Profit for A is $50, profit for B is $75
  • Write constraint inequality and graph
    • Let $x$ be number of A and $y$ be number of B
    • Constraint inequality: $2x + 3y \leq 24$
    • Graph shows possible combinations of A and B within time constraint

Linear Programming and Feasible Regions

• Linear programming involves optimizing a linear function subject to linear constraints
• Feasible region is the set of all points satisfying all constraints in a linear programming problem
• Graphing multiple linear inequalities on the same coordinate plane creates the feasible region
• The optimal solution is typically found at a vertex of the feasible region