Linear inequalities help us visualize mathematical relationships on a graph. We can plot multiple inequalities together to find areas that satisfy all conditions at once. This is super useful for solving real-world problems with multiple constraints.

Graphing these systems lets us see the solution visually. We shade areas that work for each inequality and look for where they overlap. This shaded region shows all the possible solutions that fit our requirements.

Graphing Systems of Linear Inequalities

Solutions for linear inequality systems

• A system of linear inequalities includes two or more linear inequalities with the same variables
• An ordered pair $(x, y)$ solves a system of linear inequalities if it satisfies all inequalities simultaneously
  • Substitute the $x$ and $y$ values into each inequality
  • If all inequalities are true for the given ordered pair, it is a solution to the system
• The solution set of a system of linear inequalities contains all ordered pairs that satisfy the system
  • Example: For the system $y > 2x - 1$ and $y \leq x + 3$, the ordered pair $(2, 4)$ is a solution because it satisfies both inequalities

Graphing of inequality systems

• To graph a system of linear inequalities, graph each inequality separately on the same coordinate plane
  • Convert the inequality to slope-intercept form: $y = mx + b$
  • Plot the boundary line using the slope $m$ and y-intercept $b$
    • Use a dashed line for strict inequalities $(< \text{ or } >)$ and a solid line for inclusive inequalities $(\leq \text{ or } \geq)$
  • Shade the region above the line for $y > mx + b$ or $y \geq mx + b$, and below the line for $y < mx + b$ or $y \leq mx + b$
• The solution set is the overlapping shaded region that satisfies all inequalities
• The solution region can be bounded (closed) or unbounded (open)
  • A bounded region is enclosed by the boundary lines and has a finite area (a polygon)
  • An unbounded region extends infinitely in one or more directions (open sides)

Visualizing solutions on the coordinate plane

• The coordinate plane provides a visual representation of the system's solution
• Shading is used to indicate the areas that satisfy each inequality
• The intersection of shaded regions represents the solution set for the entire system
• Constraints of the system are represented by the boundary lines of each inequality

Real-world applications of inequalities

• Systems of linear inequalities can model real-world situations with multiple constraints
• Common applications include:
  • Resource allocation problems
    • Constraints may include limited resources, time, or budget
    • Objective is to maximize or minimize a quantity while satisfying the constraints
    • Example: A company producing two products with limited raw materials and labor hours
  • Feasible region problems
    • Constraints define the boundaries of a region where a solution is possible
    • Objective is to find the optimal solution within the feasible region
    • Example: Determining the optimal production mix for maximum profit
• To solve real-world problems using systems of linear inequalities:
  1. Identify the variables and define them in terms of the problem
  2. Write inequalities to represent the constraints
  3. Graph the system of inequalities to visualize the feasible region
  4. Determine the optimal solution based on the problem's objective