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 .
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
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A includes two or more linear inequalities with the same variables
An (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 of a system of linear inequalities contains all ordered pairs that satisfy the system
Example: For the system y[>](https://www.fiveableKeyTerm:>)2x−1 and y≤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
Convert the inequality to : y=mx+b
Plot the using the slope m and y-intercept b
Use a for strict inequalities ([<](https://www.fiveableKeyTerm:<) or >) and a for inclusive inequalities (≤ or ≥)
Shade the region above the line for y>mx+b or y≥mx+b, and below the line for y<mx+b or y≤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 is enclosed by the boundary lines and has a finite area (a polygon)
An 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
is used to indicate the areas that satisfy each inequality
The 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
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:
Identify the variables and define them in terms of the problem
Write inequalities to represent the constraints
Graph the system of inequalities to visualize the feasible region
Determine the optimal solution based on the problem's objective