Solving Systems Of Linear Inequalities 60(x-2) < 3(x+50) And -7(x+3) < -5(x-3)

In the realm of mathematics, particularly in algebra, systems of linear inequalities play a crucial role in modeling real-world scenarios and finding feasible solutions. These systems involve two or more linear inequalities that must be satisfied simultaneously. Understanding how to solve these systems is essential for various applications, including optimization problems, resource allocation, and decision-making processes. This article delves into the intricacies of solving systems of linear inequalities, providing a step-by-step guide with examples to enhance comprehension. We will explore the algebraic techniques involved in isolating variables and determining the solution sets, as well as graphical methods for visualizing the solutions. Mastery of these concepts equips individuals with valuable tools for tackling complex problems across diverse fields.

Before diving into solving systems, it's vital to grasp the fundamentals of linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). These inequalities define a range of values rather than a single solution, which is characteristic of linear equations. For instance, the inequality x > 3 represents all real numbers greater than 3, while y ≤ -2 includes all real numbers less than or equal to -2. Graphically, a linear inequality in two variables represents a region in the coordinate plane, bounded by a line. The line itself is included in the solution set if the inequality is non-strict (≤ or ≥) and excluded if the inequality is strict (< or >). This graphical representation is invaluable for visualizing the solution set of a system of linear inequalities. When dealing with systems, we seek the region where the solutions of all inequalities overlap, providing a set of values that satisfy all conditions simultaneously. Understanding the individual inequalities is the cornerstone of solving the entire system effectively. This foundation allows us to approach more complex systems with confidence and precision.

The algebraic method for solving linear inequalities involves manipulating the inequalities to isolate the variable of interest. This process is similar to solving linear equations but with a critical distinction: multiplying or dividing by a negative number reverses the direction of the inequality sign. For instance, if we have -2x < 6, dividing both sides by -2 yields x > -3. This sign reversal is crucial for obtaining the correct solution set. The steps typically include simplifying the inequality by distributing and combining like terms, then performing operations on both sides to isolate the variable. For a system of inequalities, we solve each inequality separately. Consider the system:

1. 2x + y ≤ 4
2. x - y > 1

We solve each inequality for y:

1. y ≤ -2x + 4
2. y < x - 1

These inequalities now define the regions of the coordinate plane that satisfy each condition individually. The final step is to identify the intersection of these regions, which represents the solution set for the entire system. This can be done graphically or by further algebraic analysis, depending on the complexity of the system. Mastering these algebraic manipulations is essential for handling a wide range of inequality problems and lays the groundwork for more advanced mathematical techniques.

Graphical methods offer a visual approach to solving systems of linear inequalities, which can be particularly helpful for understanding the solution set. Each linear inequality represents a region in the coordinate plane, and the solution to the system is the intersection of these regions. The process begins by graphing each inequality separately. To graph a linear inequality, first, treat it as a linear equation and plot the boundary line. For inequalities with < or >, the boundary line is dashed to indicate that points on the line are not included in the solution. For ≤ or ≥, the boundary line is solid, indicating that points on the line are part of the solution. Next, shade the region that satisfies the inequality. To determine which side to shade, choose a test point (such as (0,0)) and substitute its coordinates into the inequality. If the inequality holds true, shade the side containing the test point; if not, shade the opposite side. For example, consider the inequality y > x + 1. The boundary line is y = x + 1, which is dashed. Testing the point (0,0) gives 0 > 0 + 1, which is false, so we shade the region above the line. Once all inequalities are graphed, the region where the shaded areas overlap represents the solution set of the system. This overlapping region may be a polygon, a ray, a line segment, or even an empty set, depending on the inequalities. Graphical methods provide an intuitive way to visualize solutions and are especially useful for systems with two variables. They also serve as a valuable tool for verifying solutions obtained algebraically, ensuring accuracy and understanding.

Let's apply our knowledge to solve the specific system of linear inequalities presented:

$$\begin{array}{l} 60(x-2)<3(x+50) \\ -7(x+3)<-5(x-3) \end{array}$$

First, we solve the inequality 60(x - 2) < 3(x + 50):

Distribute the constants:

60x - 120 < 3x + 150

Subtract 3x from both sides:

57x - 120 < 150

Add 120 to both sides:

57x < 270

Divide by 57:

x < 270/57

Simplify the fraction:

x < 90/19

Next, we solve the inequality -7(x + 3) < -5(x - 3):

Distribute the constants:

-7x - 21 < -5x + 15

Add 7x to both sides:

-21 < 2x + 15

Subtract 15 from both sides:

-36 < 2x

Divide by 2:

-18 < x

So, x > -18. Combining the two solutions, we have x < 90/19 and x > -18. The solution set is the interval (-18, 90/19). This means that any value of x within this range will satisfy both inequalities simultaneously. Graphically, this would be represented as the region on the number line between -18 and 90/19, excluding the endpoints. This detailed step-by-step solution demonstrates the algebraic techniques involved in solving systems of linear inequalities and highlights the importance of careful manipulation and simplification.

The solution set of a system of linear inequalities represents all the points that satisfy every inequality in the system. This set can be expressed in several ways, depending on the context and the nature of the solution. One common method is interval notation, which is particularly useful for systems with one variable. As we found in the previous example, the solution set for the system was x < 90/19 and x > -18. In interval notation, this is written as (-18, 90/19), indicating that x can take any value between -18 and 90/19, excluding the endpoints. For systems with two or more variables, the solution set is often a region in the coordinate plane. This region can be described graphically, as we discussed earlier, or using set-builder notation. For instance, if the solution region is bounded by the inequalities y ≤ -2x + 4 and y < x - 1, the solution set can be expressed as {(x, y) | y ≤ -2x + 4, y < x - 1}. This notation specifies the conditions that x and y must satisfy to be part of the solution. Another way to express the solution set is by listing the boundary lines and indicating which side of each line is included in the solution. This is particularly useful when describing the solution region geometrically. The choice of representation depends on the specific problem and the level of detail required. Understanding these different methods allows for effective communication and interpretation of solutions in various mathematical and real-world contexts.

Systems of linear inequalities have numerous applications across various fields, making their study highly relevant and practical. One significant area is in linear programming, where these systems are used to model constraints in optimization problems. For example, a company might want to maximize its profit while adhering to limitations on resources, such as labor, materials, and production time. These constraints can be expressed as linear inequalities, and the feasible region (the solution set) represents all possible production plans that satisfy these constraints. The optimal solution, which maximizes profit, can then be found within this feasible region using techniques like the simplex method or graphical analysis. Another application is in economics, where systems of inequalities can model supply and demand curves, budget constraints, and resource allocation scenarios. For instance, a consumer's budget constraint can be represented as an inequality, limiting the amount they can spend on different goods. The intersection of multiple constraints defines the consumer's feasible consumption set. In engineering, systems of inequalities are used in design and optimization problems. For example, structural engineers might use inequalities to ensure that a bridge can withstand certain loads or that a building meets safety standards. In computer science, these systems can be used in network flow problems and scheduling algorithms. Real-world applications also include nutrition planning (ensuring dietary requirements are met within calorie limits), transportation logistics (optimizing routes and delivery schedules), and environmental management (setting pollution limits). These diverse applications highlight the broad utility of systems of linear inequalities in solving practical problems and making informed decisions. Their ability to model constraints and feasible regions makes them an indispensable tool in optimization and decision-making across various disciplines.

When solving systems of linear inequalities, several common mistakes can lead to incorrect solutions. Recognizing these pitfalls and implementing strategies to avoid them is crucial for achieving accuracy. One frequent error is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. This simple oversight can completely alter the solution set. To prevent this, always double-check the sign of the multiplier or divisor and remember to flip the inequality if it's negative. Another common mistake occurs during graphical solutions: shading the wrong region. To avoid this, use a test point, as discussed earlier, to verify which side of the boundary line should be shaded. A test point not on the line is substituted into the inequality; if it satisfies the inequality, shade the region containing the test point; otherwise, shade the other region. Misinterpreting the boundary line is also a common error. Remember that a dashed line indicates the points on the line are not included in the solution (strict inequalities < or >), while a solid line means they are included (non-strict inequalities ≤ or ≥). Sloppy algebraic manipulations, such as incorrect distribution or combining like terms, can also lead to mistakes. Take your time, write each step clearly, and double-check your work. When dealing with word problems, a frequent error is misinterpreting the problem's constraints and translating them into incorrect inequalities. Carefully read the problem, identify the key variables and relationships, and translate them accurately into mathematical statements. Finally, failing to check the solution is a significant oversight. After solving the system, substitute the solution back into the original inequalities to ensure it satisfies all conditions. By being mindful of these common mistakes and implementing strategies to avoid them, you can significantly improve your accuracy and confidence in solving systems of linear inequalities.

In conclusion, solving systems of linear inequalities is a fundamental skill in mathematics with broad applications across various fields. This article has provided a comprehensive guide to understanding and solving these systems, covering algebraic and graphical methods, expressing the solution set, and highlighting common mistakes to avoid. We began by defining linear inequalities and emphasizing their role in representing a range of values. We then explored algebraic techniques for isolating variables and manipulating inequalities, stressing the critical rule of reversing the inequality sign when multiplying or dividing by a negative number. Graphical methods were presented as a visual aid, allowing for the representation of inequalities as regions in the coordinate plane and the identification of the solution set as the intersection of these regions. A step-by-step solution to a specific system demonstrated the practical application of these techniques. Furthermore, we discussed various ways to express the solution set, including interval notation and set-builder notation, and highlighted the importance of choosing the appropriate representation for the context. The numerous applications of systems of linear inequalities, from linear programming to economics and engineering, underscore their relevance in solving real-world problems. Finally, we addressed common mistakes and provided strategies to avoid them, emphasizing the importance of careful manipulation, accurate graphing, and thorough checking of solutions. Mastery of these concepts equips individuals with valuable tools for tackling complex problems and making informed decisions in various mathematical and real-world scenarios.