Solving 60(x-2)<3(x+50) And -7(x+3)<-5(x-3) A Step-by-Step Guide

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Introduction to Inequalities and Systems of Inequalities

In mathematics, inequalities play a crucial role in defining relationships where values are not necessarily equal. Understanding inequalities is essential for solving a wide array of problems, from determining the range of possible solutions in real-world scenarios to optimizing mathematical models. An 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). Unlike equations, which have a single solution or a set of discrete solutions, inequalities often have a range of solutions. This range can be represented graphically on a number line or as an interval.

A system of inequalities, on the other hand, involves two or more inequalities that must be solved simultaneously. The solution to a system of inequalities is the set of values that satisfy all the inequalities in the system. These systems are frequently encountered in various fields, including economics, engineering, and computer science, where constraints and limitations need to be considered collectively. Solving a system of inequalities requires finding the intersection of the solution sets of the individual inequalities. This means identifying the values that make all the inequalities true at the same time. The solution set can be represented graphically as the region where the shaded areas of the individual inequalities overlap. Analytically, solving such systems involves algebraic manipulations to isolate the variable in each inequality and then combining the results to find the common solution range. Understanding how to manipulate and solve inequalities is fundamental to tackling more complex mathematical problems and real-world applications.

To effectively solve systems of inequalities, it is important to have a solid grasp of basic algebraic principles. This includes the ability to simplify expressions, combine like terms, and apply the distributive property. Additionally, understanding the rules for manipulating inequalities is crucial. For example, multiplying or dividing both sides of an inequality by a negative number requires reversing the inequality sign. This is a critical step that, if overlooked, can lead to an incorrect solution. Furthermore, the ability to interpret and represent solutions graphically is a valuable skill. Visualizing the solution set can provide a clear understanding of the range of possible values that satisfy the system of inequalities. In this comprehensive guide, we will delve into the step-by-step process of solving the specific system of inequalities: 60(x-2)<3(x+50) and -7(x+3)<-5(x-3), providing detailed explanations and strategies to master this topic. Whether you are a student learning algebra or a professional needing to apply these concepts in your field, this guide aims to equip you with the necessary tools and knowledge to confidently solve systems of inequalities.

Step-by-Step Solution of 60(x-2)<3(x+50)

The first inequality we need to solve is 60(x-2)<3(x+50). This inequality involves a variable x and requires us to isolate x to find the range of values that satisfy the condition. The process begins by applying the distributive property to both sides of the inequality. The distributive property states that a(b + c) = ab + ac, which allows us to remove the parentheses and simplify the expression.

  1. Apply the distributive property:

    • Distribute the 60 on the left side: 60 * x - 60 * 2, which simplifies to 60x - 120.
    • Distribute the 3 on the right side: 3 * x + 3 * 50, which simplifies to 3x + 150.

    The inequality now looks like this: 60x - 120 < 3x + 150.

  2. Collect like terms:

    • To isolate the terms with x on one side, subtract 3x from both sides of the inequality. This gives us: 60x - 3x - 120 < 3x - 3x + 150, which simplifies to 57x - 120 < 150.
    • Next, we want to isolate the constant terms on the other side. Add 120 to both sides of the inequality: 57x - 120 + 120 < 150 + 120, which simplifies to 57x < 270.

  3. Isolate the variable x:

    • To solve for x, divide both sides of the inequality by 57: (57x) / 57 < 270 / 57.
    • This simplifies to x < 270/57.

  4. Simplify the fraction:

    • The fraction 270/57 can be simplified by finding the greatest common divisor (GCD) of 270 and 57. Both numbers are divisible by 3, so we can simplify: 270 ÷ 3 = 90 and 57 ÷ 3 = 19.
    • Thus, the simplified fraction is 90/19.

  5. Final solution for the first inequality:

    • The solution for the first inequality is x < 90/19. This means that any value of x less than 90/19 will satisfy the inequality 60(x-2)<3(x+50).

In summary, solving the inequality 60(x-2)<3(x+50) involves applying the distributive property, collecting like terms, and isolating the variable x. The final solution, x < 90/19, represents the range of values for x that make the inequality true. This step-by-step approach ensures clarity and accuracy in the solution process. Understanding these fundamental steps is crucial for solving more complex inequalities and systems of inequalities. The ability to confidently manipulate algebraic expressions and apply the rules of inequalities is a valuable skill in mathematics and various fields that rely on mathematical modeling.

Step-by-Step Solution of -7(x+3)<-5(x-3)

Now, let's tackle the second inequality in the system: -7(x+3)<-5(x-3). This inequality, like the first, requires us to isolate the variable x to determine the range of values that satisfy the condition. We'll follow a similar process, starting with the distributive property.

  1. Apply the distributive property:

    • Distribute the -7 on the left side: -7 * x + (-7) * 3, which simplifies to -7x - 21.
    • Distribute the -5 on the right side: -5 * x + (-5) * (-3), which simplifies to -5x + 15.

    The inequality now looks like this: -7x - 21 < -5x + 15.

  2. Collect like terms:

    • To gather the terms with x on one side, add 5x to both sides of the inequality: -7x + 5x - 21 < -5x + 5x + 15, which simplifies to -2x - 21 < 15.
    • Next, isolate the constant terms on the other side by adding 21 to both sides: -2x - 21 + 21 < 15 + 21, which simplifies to -2x < 36.

  3. Isolate the variable x:

    • To solve for x, divide both sides of the inequality by -2. Remember, when dividing (or multiplying) both sides of an inequality by a negative number, you must reverse the direction of the inequality sign: (-2x) / -2 > 36 / -2.
    • This simplifies to x > -18.

  4. Final solution for the second inequality:

    • The solution for the second inequality is x > -18. This means that any value of x greater than -18 will satisfy the inequality -7(x+3)<-5(x-3).

In summary, the process of solving the inequality -7(x+3)<-5(x-3) involves distributing, collecting like terms, and isolating the variable x. A critical step is remembering to reverse the inequality sign when dividing by a negative number. The final solution, x > -18, indicates the range of x values that make the inequality true. This methodical approach is essential for accuracy in solving inequalities. Grasping these steps and rules enables one to tackle a wider range of mathematical problems, especially those involving constraints and boundaries. The ability to manipulate and solve inequalities is a foundational skill that supports advanced mathematical concepts and practical applications in various disciplines.

Combining the Solutions and Finding the Intersection

After solving the individual inequalities, 60(x-2)<3(x+50) and -7(x+3)<-5(x-3), we found the solutions to be x < 90/19 and x > -18, respectively. Now, the critical step is to combine these solutions to find the intersection, which represents the set of values that satisfy both inequalities simultaneously. This intersection is the solution to the system of inequalities.

  1. Represent the solutions on a number line:

    • Draw a number line and mark the key points: -18 and 90/19. Note that 90/19 is approximately 4.74, so it lies between 4 and 5 on the number line.
    • For the inequality x < 90/19, shade the region to the left of 90/19. Since the inequality is strictly less than, use an open circle at 90/19 to indicate that this point is not included in the solution.
    • For the inequality x > -18, shade the region to the right of -18. Similarly, use an open circle at -18 to indicate that this point is not included in the solution.

  2. Identify the overlapping region:

    • The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap. In this case, the overlapping region is between -18 and 90/19. This means that the values of x that satisfy both inequalities are those that are greater than -18 and less than 90/19.

  3. Express the solution in interval notation:

    • Interval notation is a concise way to represent the solution set. For the overlapping region between -18 and 90/19, the interval notation is (-18, 90/19). The parentheses indicate that the endpoints -18 and 90/19 are not included in the solution, which aligns with the open circles used on the number line.

  4. Write the solution as a compound inequality:

    • Alternatively, the solution can be expressed as a compound inequality: -18 < x < 90/19. This notation explicitly states that x must be greater than -18 and less than 90/19.

In summary, combining the solutions to the individual inequalities involves identifying the range of values that satisfy both conditions. Representing the solutions on a number line is a useful visual aid to find the overlapping region. The solution to the system of inequalities 60(x-2)<3(x+50) and -7(x+3)<-5(x-3) is the interval (-18, 90/19), or equivalently, the compound inequality -18 < x < 90/19. This comprehensive approach ensures a clear understanding of how to solve systems of inequalities and interpret the results. The ability to combine solutions and express them in various notations is a crucial skill in mathematics and is applicable in many real-world scenarios involving constraints and conditions.

Graphical Representation of the Solution

Visualizing the solution to a system of inequalities through graphical representation is a powerful method for enhancing understanding. Graphing the solution not only confirms the analytical results but also provides a clear picture of the range of values that satisfy all inequalities simultaneously. This graphical approach is particularly useful when dealing with more complex systems or when there are multiple variables involved. For the system of inequalities we solved, 60(x-2)<3(x+50) and -7(x+3)<-5(x-3), the graphical representation involves plotting each inequality on a number line and identifying the region where their solutions overlap.

  1. Representing the first inequality graphically:

    • The solution to the first inequality, x < 90/19, can be represented on a number line by shading the region to the left of 90/19. Since the inequality is strict (less than), we use an open circle at 90/19 to indicate that this point is not included in the solution.

  2. Representing the second inequality graphically:

    • Similarly, the solution to the second inequality, x > -18, is represented by shading the region to the right of -18 on the number line. Again, since the inequality is strict (greater than), we use an open circle at -18.

  3. Identifying the overlapping region:

    • The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. Visually, this is the section of the number line where both shadings are present. In our case, this is the region between -18 and 90/19.

  4. Interpreting the graph:

    • The graph clearly shows that the solution set includes all values of x that are greater than -18 and less than 90/19. The open circles at -18 and 90/19 reinforce that these specific values are not part of the solution set. This visual representation aligns perfectly with the analytical solution we derived, which is the interval (-18, 90/19) or the compound inequality -18 < x < 90/19.

Graphical representation is not only a method to confirm solutions but also a valuable tool for understanding the nature of inequalities. It helps in visualizing the range of possible solutions and the boundaries that define these ranges. In more complex systems involving multiple inequalities or variables, graphing can provide insights that algebraic methods might not reveal as readily. For instance, in two-variable systems, graphing the inequalities on a coordinate plane helps identify the feasible region, which represents all points that satisfy all the inequalities. This is a fundamental concept in linear programming and optimization problems. The ability to translate algebraic solutions into graphical representations and vice versa is a critical skill in mathematics and related fields. It fosters a deeper understanding of the concepts and enhances problem-solving capabilities. In summary, the graphical representation of the solution to the system of inequalities 60(x-2)<3(x+50) and -7(x+3)<-5(x-3) provides a clear and intuitive understanding of the solution set, reinforcing the analytical results and demonstrating the power of visual methods in mathematics.

Real-World Applications of Systems of Inequalities

Systems of inequalities are not just abstract mathematical concepts; they have numerous real-world applications across various fields, making them a crucial tool for problem-solving and decision-making. Real-world applications of these systems often involve scenarios where multiple constraints or conditions need to be satisfied simultaneously. Understanding how to model and solve these systems is essential for making informed decisions in fields ranging from economics and engineering to computer science and operations research.

  1. Economics and Finance:

    • In economics, systems of inequalities are used to model budget constraints and resource allocation. For example, a consumer might have a limited budget and need to decide how to allocate it between different goods and services. The constraints on spending and the prices of the goods can be represented as a system of inequalities. The solution to this system provides the range of feasible consumption bundles that the consumer can afford.
    • In finance, portfolio optimization involves allocating investments across different assets to maximize returns while minimizing risk. This often involves setting constraints on the amount invested in each asset, the overall risk level, and the desired return. These constraints can be formulated as a system of inequalities, and the solution provides the optimal investment strategy.

  2. Engineering:

    • In engineering, systems of inequalities are used in design and optimization problems. For example, in structural engineering, the design of a bridge must satisfy constraints on the load it can bear, the materials used, and the cost. These constraints can be expressed as inequalities, and the solution provides the range of feasible designs that meet the required specifications.
    • In control systems, inequalities are used to define stability criteria and performance requirements. The system must operate within certain bounds to ensure stability and meet the desired performance targets. These conditions can be formulated as inequalities, and the solution helps engineers design controllers that satisfy these requirements.

  3. Operations Research:

    • Operations research is a field that uses mathematical models to solve complex decision-making problems. Linear programming, a core technique in operations research, heavily relies on systems of inequalities. Linear programming problems involve optimizing a linear objective function subject to linear inequality constraints.
    • For example, a manufacturing company might want to maximize its profit by determining the optimal production levels for different products. The constraints might include limitations on resources such as raw materials, labor, and machine time. These constraints can be formulated as a system of inequalities, and the solution provides the production plan that maximizes profit while satisfying all constraints.

  4. Computer Science:

    • In computer science, systems of inequalities are used in areas such as network flow optimization and resource allocation in distributed systems. For example, in a network, the flow of data must satisfy capacity constraints on the links. These constraints can be represented as inequalities, and the solution provides the maximum flow that can be supported by the network.

In summary, systems of inequalities provide a versatile framework for modeling and solving problems with multiple constraints. The real-world applications span various disciplines, highlighting the importance of understanding these mathematical concepts. From optimizing financial portfolios and engineering designs to managing supply chains and network resources, systems of inequalities offer valuable tools for decision-making and problem-solving. The ability to formulate real-world problems as systems of inequalities and solve them effectively is a crucial skill for professionals in many fields. The solutions to these systems not only provide feasible ranges but also enable informed choices that lead to optimal outcomes.

Conclusion: Mastering Systems of Inequalities

In conclusion, mastering the solution of systems of inequalities is a fundamental skill in mathematics with wide-ranging applications across various disciplines. Throughout this comprehensive guide, we have explored the step-by-step process of solving the system 60(x-2)<3(x+50) and -7(x+3)<-5(x-3), illustrating the key techniques and principles involved. From applying the distributive property and collecting like terms to isolating the variable and combining solutions, each step is crucial for arriving at the correct answer. The ability to accurately manipulate inequalities and interpret their solutions is essential for tackling more complex mathematical problems and real-world applications.

We began by introducing the concept of inequalities and systems of inequalities, emphasizing their significance in mathematical problem-solving and modeling real-world scenarios. Understanding the symbols and rules governing inequalities sets the foundation for solving systems effectively. We then delved into the detailed solution of each inequality individually, demonstrating the algebraic manipulations required to isolate the variable. This involved applying the distributive property to remove parentheses, collecting like terms to simplify the expressions, and performing operations on both sides of the inequality to isolate the variable. A critical step highlighted was the importance of reversing the inequality sign when multiplying or dividing by a negative number, a common source of errors if overlooked.

Next, we focused on combining the solutions of the individual inequalities to find the intersection, which represents the solution set for the system. This involved representing the solutions on a number line, identifying the overlapping region, and expressing the solution in both interval notation and as a compound inequality. This process reinforces the understanding that the solution to a system of inequalities is the set of values that satisfy all inequalities simultaneously. The graphical representation of the solutions further enhanced comprehension, providing a visual confirmation of the analytical results and illustrating the range of possible values.

Furthermore, we explored the real-world applications of systems of inequalities across various fields, including economics, engineering, operations research, and computer science. These examples demonstrated the practical relevance of systems of inequalities in solving optimization problems, resource allocation, and decision-making under constraints. The ability to translate real-world scenarios into mathematical models involving systems of inequalities is a valuable skill for professionals in many domains.

By mastering the techniques presented in this guide, you can confidently approach and solve a wide range of problems involving systems of inequalities. The key is to practice and apply these methods consistently, developing a strong understanding of the underlying principles. Whether you are a student learning algebra or a professional applying mathematical models in your field, the ability to solve systems of inequalities is an invaluable asset. This comprehensive guide aims to equip you with the necessary knowledge and skills to succeed in this area, enabling you to tackle complex problems and make informed decisions based on mathematical analysis. Mastering systems of inequalities not only enhances your mathematical proficiency but also empowers you to address real-world challenges with greater confidence and effectiveness.