Solving Inequalities If 5(4-x) Greater Than 3x+1

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Introduction

In this comprehensive guide, we will delve into the world of inequalities, focusing on the specific problem: If 5(4-x) > 3x + 1, which of the following is correct? We will explore the fundamental concepts of inequalities, discuss various methods for solving them, and provide step-by-step solutions to the given problem. Understanding inequalities is crucial in various fields, including mathematics, physics, economics, and computer science. This guide aims to equip you with the knowledge and skills necessary to confidently tackle inequality problems.

Inequalities are mathematical statements that compare two expressions using symbols such as >, <, ≥, and ≤. Unlike equations, which state that two expressions are equal, inequalities indicate that one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. Solving inequalities involves finding the range of values that satisfy the given inequality. This often requires manipulating the inequality while preserving its truth value. The basic principles of algebra, such as addition, subtraction, multiplication, and division, apply to inequalities with some key differences. For instance, multiplying or dividing an inequality by a negative number reverses the inequality sign.

This guide will not only provide the solution to the specific problem but also offer a broader understanding of how to approach and solve inequalities. We will cover the essential properties of inequalities, demonstrate step-by-step solution strategies, and provide examples to illustrate the concepts. By the end of this guide, you will have a solid foundation in solving inequalities and be well-prepared to tackle more complex problems. Let's begin our journey into the world of inequalities and unravel the mysteries behind them. Remember, understanding the core principles and practicing consistently are key to mastering this fundamental mathematical concept.

Understanding Inequalities

Understanding Inequalities: Before diving into solving the specific problem, let's establish a solid foundation in the basics of inequalities. Inequalities are mathematical statements that compare two expressions using symbols such as >, <, ≥, and ≤. These symbols represent 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to', respectively. Unlike equations, which seek to find exact values that make two expressions equal, inequalities define a range of values that satisfy the given condition. This range can be finite or infinite, depending on the nature of the inequality.

To effectively work with inequalities, it's essential to grasp some key concepts. Firstly, the properties of inequalities dictate how we can manipulate them without changing the solution set. For example, adding or subtracting the same number from both sides of an inequality preserves the inequality. Similarly, multiplying or dividing both sides by a positive number maintains the inequality. However, a crucial rule to remember is that multiplying or dividing by a negative number reverses the inequality sign. This is a fundamental principle that must be applied carefully to avoid errors.

Another important concept is the graphical representation of inequalities. Inequalities can be represented on a number line, which visually illustrates the range of values that satisfy the inequality. For example, x > 3 would be represented by an open circle at 3 (indicating that 3 is not included) and a line extending to the right, representing all values greater than 3. Similarly, x ≤ 5 would be represented by a closed circle at 5 (indicating that 5 is included) and a line extending to the left, representing all values less than or equal to 5. Understanding these graphical representations can provide a more intuitive grasp of the solutions to inequalities.

Moreover, compound inequalities, which combine two or more inequalities, require special attention. These can be in the form of 'and' or 'or' statements. An 'and' statement means that both inequalities must be satisfied simultaneously, while an 'or' statement means that at least one of the inequalities must be satisfied. Solving compound inequalities involves finding the intersection or union of the solution sets of the individual inequalities, depending on whether it's an 'and' or 'or' statement.

In summary, a thorough understanding of the properties, graphical representations, and types of inequalities is crucial for solving them effectively. These foundational concepts will serve as the building blocks for tackling more complex inequality problems, including the one we will address in the following sections. By mastering these basics, you'll be well-equipped to navigate the world of inequalities with confidence and precision.

Problem Statement and Analysis

Problem Statement and Analysis: Let's revisit the problem at hand: If 5(4-x) > 3x + 1, which of the following options is correct? This problem presents a linear inequality, which is an inequality that involves a linear expression. To solve this, we need to isolate the variable 'x' on one side of the inequality. This involves applying algebraic manipulations while adhering to the rules of inequalities.

Before we dive into the step-by-step solution, it's beneficial to analyze the structure of the inequality. The left-hand side of the inequality, 5(4-x), involves a distributive property application. This means we need to multiply 5 by both terms inside the parentheses, which are 4 and -x. The right-hand side, 3x + 1, is a simple linear expression. Our goal is to simplify both sides and then rearrange the terms to get 'x' by itself.

To approach this problem methodically, we should first distribute the 5 on the left-hand side. This will give us 20 - 5x. The inequality then becomes 20 - 5x > 3x + 1. Next, we need to collect the 'x' terms on one side and the constants on the other side. A common strategy is to add 5x to both sides to eliminate the negative coefficient on the 'x' term on the left. This results in 20 > 8x + 1. Now, we can subtract 1 from both sides to isolate the term with 'x', which gives us 19 > 8x.

Finally, to solve for 'x', we divide both sides by 8. Since 8 is a positive number, we do not need to reverse the inequality sign. This gives us 19/8 > x, which can also be written as x < 19/8. This is the solution to the inequality. However, the problem asks us to identify the correct option from a set of choices, which likely involves comparing this solution with other expressions or inequalities.

Therefore, the next step is to carefully examine the given options and compare them with our derived solution. This might involve simplifying the options or rewriting them in a similar form to our solution (x < 19/8). By systematically analyzing the problem and applying the correct algebraic manipulations, we can confidently arrive at the correct answer. The key is to remain organized, double-check each step, and ensure that we are correctly applying the rules of inequalities. This analytical approach will not only help us solve this particular problem but also equip us with the skills to tackle a wide range of inequality problems.

Step-by-Step Solution

Step-by-Step Solution: Now, let's walk through the detailed solution to the inequality 5(4-x) > 3x + 1. This step-by-step approach will ensure clarity and accuracy in our solution.

  1. Distribute the 5 on the left-hand side: The first step is to apply the distributive property to the left-hand side of the inequality. This involves multiplying 5 by both terms inside the parentheses: 5 * 4 and 5 * -x. This gives us 20 - 5x.

    • 5(4 - x) > 3x + 1
    • 20 - 5x > 3x + 1

  2. Collect 'x' terms on one side: To isolate 'x', we need to gather all the terms containing 'x' on one side of the inequality. A common strategy is to add 5x to both sides to eliminate the negative coefficient on the 'x' term on the left. This ensures we are working with positive coefficients whenever possible, reducing the chance of errors.

    • 20 - 5x + 5x > 3x + 1 + 5x
    • 20 > 8x + 1

  3. Isolate the term with 'x': Next, we need to isolate the term with 'x' by subtracting 1 from both sides of the inequality. This will move the constant term from the right-hand side to the left-hand side.

    • 20 - 1 > 8x + 1 - 1
    • 19 > 8x

  4. Solve for 'x': Finally, to solve for 'x', we divide both sides of the inequality by 8. Since 8 is a positive number, we do not need to reverse the inequality sign. This gives us the solution for 'x'.

    • 19 / 8 > 8x / 8
    • 19/8 > x

  5. Rewrite the solution: It's often helpful to rewrite the solution with 'x' on the left-hand side. This simply involves flipping the inequality and the expression, ensuring the inequality sign points in the correct direction.

    • x < 19/8

Thus, the solution to the inequality 5(4-x) > 3x + 1 is x < 19/8. This means that any value of 'x' less than 19/8 will satisfy the original inequality. Now, we need to compare this solution with the given options to determine the correct answer.

By following these steps meticulously, we have arrived at the solution in a clear and organized manner. Each step is justified by the properties of inequalities and algebraic manipulations. This systematic approach not only helps us solve the problem accurately but also enhances our understanding of the process involved in solving inequalities. In the next section, we will compare this solution with the given options to identify the correct answer.

Comparing with Options and Identifying the Correct Answer

Comparing with Options and Identifying the Correct Answer: Now that we have derived the solution x < 19/8 for the inequality 5(4-x) > 3x + 1, the next crucial step is to compare this solution with the given options to identify the correct answer. This involves carefully examining each option and determining if it accurately represents or is equivalent to our solution.

The options provided are:

A. 3x + 1 - 5(4-x) = y + 12 B. 5(4-x) + 3x + 1 = y + 12 C. 5(4-x) < 3x + 1 D. 3x + 1 < 5(4+x)

Let's analyze each option:

  • Option A: 3x + 1 - 5(4-x) = y + 12: This option is an equation, not an inequality, and it introduces a new variable 'y'. It does not directly relate to our original inequality or its solution. Therefore, this option is not correct.

  • Option B: 5(4-x) + 3x + 1 = y + 12: Similar to option A, this is also an equation involving a new variable 'y'. It does not represent an inequality and therefore cannot be the correct answer.

  • Option C: 5(4-x) < 3x + 1: This option presents an inequality, but it's the reverse of our original inequality 5(4-x) > 3x + 1. Therefore, its solution will be the opposite of our solution. Solving this inequality would lead to x > 19/8, which is not what we found. Thus, this option is incorrect.

  • Option D: 3x + 1 < 5(4+x): This option is an inequality, and it's the most likely candidate to be related to our solution. To verify, let's solve this inequality step-by-step:

    1. Distribute the 5 on the right-hand side: 3x + 1 < 20 + 5x
    2. Subtract 3x from both sides: 1 < 20 + 2x
    3. Subtract 20 from both sides: -19 < 2x
    4. Divide both sides by 2: -19/2 < x
    5. Rewrite the solution: x > -19/2

This solution, x > -19/2, is different from our derived solution x < 19/8. Therefore, option D is also incorrect.

Upon closer examination, it seems there might be an issue with the provided options, as none of them directly match or are equivalent to our solution x < 19/8. It's possible that there was a mistake in the options provided or in the problem statement itself. In such cases, it's essential to double-check the original problem and the options for any errors. If no errors are found, it might be necessary to conclude that none of the given options is correct.

In summary, by systematically comparing our solution with each option, we have determined that none of the options accurately represent the solution to the original inequality. This highlights the importance of careful analysis and verification in problem-solving. While we have not found a correct answer among the given options, we have demonstrated a thorough approach to solving inequalities and comparing solutions.

Conclusion

Conclusion: In this comprehensive guide, we have explored the intricacies of solving inequalities, focusing on the specific problem: If 5(4-x) > 3x + 1, which of the following is correct? We began by establishing a solid foundation in the basics of inequalities, including their properties, graphical representations, and different types. We then delved into the step-by-step solution of the given inequality, meticulously applying algebraic manipulations while adhering to the rules of inequalities. Our solution led us to the result x < 19/8.

Following the solution, we compared our result with the provided options, carefully analyzing each one to determine if it matched or was equivalent to our solution. Through this process, we discovered that none of the given options accurately represented the solution x < 19/8. This could indicate a potential error in the options themselves or in the original problem statement. In such situations, it's crucial to double-check all aspects of the problem to ensure accuracy.

Despite not finding a correct answer among the given options, this exercise has provided valuable insights into the process of solving inequalities and verifying solutions. We have reinforced the importance of a systematic approach, careful algebraic manipulation, and thorough comparison with potential answers. These skills are essential not only for solving mathematical problems but also for critical thinking and problem-solving in various fields.

Throughout this guide, we have emphasized the significance of understanding the fundamental concepts of inequalities, such as the properties that govern their manipulation and the impact of multiplying or dividing by negative numbers. We have also highlighted the importance of graphical representations in visualizing solutions and gaining a deeper understanding of inequalities.

In conclusion, while we did not identify a correct answer from the given options, we have successfully demonstrated the process of solving inequalities and verifying solutions. This guide serves as a valuable resource for anyone seeking to enhance their understanding of inequalities and improve their problem-solving skills. Remember, the key to mastering inequalities lies in consistent practice and a solid grasp of the underlying principles. Continue to explore and challenge yourself with different types of inequality problems to further develop your expertise in this area.