Simultaneous inequalities are systems of two or more inequalities that must all be satisfied at the same time. Unlike single inequalities, we need to find values that work for every inequality in the system. The general form includes multiple inequalities with the same variables. The solution is the intersection of all individual solution sets, representing values that satisfy all constraints simultaneously. This concept is fundamental in optimization, linear programming, and many real-world applications where multiple conditions must be met.
Let's solve a concrete example of simultaneous linear inequalities. We have the system: 2x plus 3 greater than 7, and x minus 1 less than or equal to 4. First, we solve each inequality separately. For the first inequality, subtract 3 from both sides to get 2x greater than 4, then divide by 2 to get x greater than 2. For the second inequality, add 1 to both sides to get x less than or equal to 5. Now we find the intersection of these solutions. We need x greater than 2 AND x less than or equal to 5. This gives us the solution 2 less than x less than or equal to 5, written in interval notation as open parenthesis 2 comma 5 close bracket.
Now let's explore the graphical method for solving simultaneous inequalities. Consider the system: x plus y less than or equal to 4, and x minus y greater than or equal to 0. In the graphical method, each inequality represents a half-plane on the coordinate system. The first inequality, x plus y less than or equal to 4, represents the region below and on the line x plus y equals 4. The second inequality, x minus y greater than or equal to 0, represents the region above and on the line x minus y equals 0. The solution to the system is the intersection of these regions, called the feasible region. This shaded area contains all points that satisfy both inequalities simultaneously. The graphical method provides visual confirmation of our algebraic solutions.
Let's work through a more complex example with three simultaneous inequalities: x plus y less than or equal to 6, 2x minus y greater than or equal to 2, and x greater than or equal to 0. Algebraically, we can rewrite these as y less than or equal to 6 minus x, y less than or equal to 2x minus 2, and x greater than or equal to 0. Graphically, each inequality creates a half-plane, and their intersection forms a triangular feasible region. The vertices of this region are at (0,0), (1,0), and (8/3, 10/3). We can verify our solution by testing a point like (2,2): it satisfies 2 plus 2 equals 4 which is less than 6, 2 times 2 minus 2 equals 2 which satisfies the second inequality, and 2 is greater than 0. This systematic approach works for any number of linear inequalities.
Let's explore a real-world application of simultaneous inequalities through a production optimization problem. A furniture factory produces chairs and tables with limited resources. The constraints are: material constraint where 2 units of wood per chair plus 4 units per table cannot exceed 100 units total, and a labor constraint where 3 hours per chair plus 2 hours per table cannot exceed 90 hours. Additionally, production quantities must be non-negative. These real-world constraints translate directly into our mathematical inequalities: 2x plus 4y less than or equal to 100, 3x plus 2y less than or equal to 90, x greater than or equal to 0, and y greater than or equal to 0. The feasible region shows all possible production combinations, with vertices representing potential optimal solutions. In practice, businesses use this framework to maximize profit or minimize cost within their operational constraints.