Solving Inequalities Finding Solutions For 47 < -5-8v

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In the realm of mathematics, inequalities play a pivotal role in defining relationships between values that are not necessarily equal. Unlike equations that assert equality, inequalities express a range of possible values, making them indispensable in various fields, from economics to engineering. This article delves into the intricacies of solving inequalities, focusing on the specific example of 47 < -5-8v. We will dissect the methods to identify solutions, understand the underlying principles, and apply these concepts to real-world scenarios. Our primary focus will be on finding the values of 'v' that satisfy the given inequality. By the end of this comprehensive guide, you will be equipped with the skills to tackle similar problems and gain a deeper appreciation for the power of inequalities in mathematical problem-solving.

Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which have specific solutions, inequalities often have a range of solutions. Understanding how to solve inequalities is crucial in various fields, including economics, physics, and computer science. Solving inequalities involves isolating the variable on one side of the inequality sign while maintaining the truth of the statement. This often requires performing the same operations on both sides, with one key difference compared to equations: multiplying or dividing by a negative number reverses the inequality sign. This article will explore the step-by-step process of solving the inequality 47 < -5 - 8v, and we will also verify whether specific values satisfy the inequality.

Before diving into the solution, let's break down the inequality 47 < -5-8v. This statement reads as "47 is less than -5 minus 8 times v." Our goal is to find all values of 'v' that make this statement true. The process involves isolating 'v' on one side of the inequality. We will employ algebraic manipulations, ensuring we maintain the balance of the inequality. The key here is to remember that multiplying or dividing by a negative number will flip the inequality sign. This is a crucial step to keep in mind as we work through the problem. By carefully applying the rules of algebra, we can systematically narrow down the possible values of 'v' until we arrive at the solution set. Understanding the structure of the inequality is the first step towards successfully solving it and applying the solution in various contexts.

The given inequality, 47 < -5-8v, is a linear inequality involving a single variable, 'v'. Linear inequalities are similar to linear equations, but instead of an equals sign, they use inequality symbols. To solve this inequality, we aim to isolate 'v' on one side. This involves performing operations on both sides of the inequality while maintaining its validity. The most important rule to remember when solving inequalities is that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. In this specific case, we need to consider how the term '-8v' affects the inequality. The negative coefficient of 'v' will require us to flip the inequality sign at some point during the solution process. Therefore, understanding this principle is essential for finding the correct solution set for 'v'.

To solve the inequality 47 < -5-8v, we'll follow a step-by-step approach to isolate the variable 'v'.

  1. Isolate the term with 'v':

    • First, we need to get the term containing 'v' by itself on one side of the inequality. We can do this by adding 5 to both sides of the inequality:

      47 + 5 < -5 - 8v + 5
52 < -8v

  2. Divide by the coefficient of 'v':

    • Next, we need to divide both sides by the coefficient of 'v', which is -8. Remember, since we are dividing by a negative number, we must reverse the inequality sign:

      52 / -8 > -8v / -8
-6.5 > v

  3. Rewrite the inequality:

    • It's often clearer to write the inequality with the variable on the left side. So, we rewrite -6.5 > v as:

      v < -6.5

This means that any value of 'v' less than -6.5 will satisfy the original inequality. Now, let's verify this solution and test the given values.

To ensure our solution is correct, we need to verify it. Our solution states that v < -6.5. This means any value of 'v' less than -6.5 should satisfy the original inequality 47 < -5-8v. We will now test the given values: v = -10, v = 7, v = 2, and v = -12, to see which ones make the inequality true. This process involves substituting each value of 'v' into the original inequality and checking if the resulting statement is true. Verification is a crucial step in solving inequalities as it helps to identify any errors made during the algebraic manipulation process. This step will also provide a clear understanding of which of the given options are indeed solutions to the inequality.

  1. Test v = -10:

    • Substitute -10 for 'v' in the original inequality:

      47 < -5 - 8(-10)
47 < -5 + 80
47 < 75

    • This statement is true, so v = -10 is a solution.

  2. Test v = 7:

    • Substitute 7 for 'v' in the original inequality:

      47 < -5 - 8(7)
47 < -5 - 56
47 < -61

    • This statement is false, so v = 7 is not a solution.

  3. Test v = 2:

    • Substitute 2 for 'v' in the original inequality:

      47 < -5 - 8(2)
47 < -5 - 16
47 < -21

    • This statement is false, so v = 2 is not a solution.

  4. Test v = -12:

    • Substitute -12 for 'v' in the original inequality:

      47 < -5 - 8(-12)
47 < -5 + 96
47 < 91

    • This statement is true, so v = -12 is a solution.

From these tests, we can see that v = -10 and v = -12 are solutions to the inequality, while v = 7 and v = 2 are not.

After verifying the solutions, it is evident that v = -10 and v = -12 are the values that satisfy the inequality 47 < -5-8v. These values fall within the solution set v < -6.5, which we derived algebraically. On the other hand, v = 7 and v = 2 do not satisfy the inequality, as they are greater than -6.5. This exercise highlights the importance of not only solving inequalities algebraically but also verifying the solutions by substituting them back into the original inequality. This ensures the accuracy of the solution and reinforces the understanding of the inequality's solution set. The process of identifying solutions involves a combination of algebraic manipulation and verification, both of which are essential skills in mathematics.

Based on our step-by-step solution and verification, the values v = -10 and v = -12 are solutions to the inequality 47 < -5-8v. When we substitute these values into the inequality, we obtain true statements. Specifically, for v = -10, the inequality becomes 47 < 75, which is true. Similarly, for v = -12, the inequality becomes 47 < 91, which is also true. This confirms that our algebraic solution of v < -6.5 is correct. The values v = 7 and v = 2, on the other hand, do not satisfy the inequality, as they result in false statements when substituted. This comprehensive approach, combining algebraic manipulation and direct verification, ensures we accurately identify the solutions to the given inequality.

In conclusion, solving inequalities is a fundamental skill in mathematics with wide-ranging applications. By understanding the principles of inequality manipulation and the importance of reversing the inequality sign when multiplying or dividing by a negative number, we can effectively find the solution sets for various inequalities. In the case of 47 < -5-8v, we demonstrated the step-by-step process of isolating the variable 'v' and arriving at the solution v < -6.5. We further validated this solution by testing the given values, confirming that v = -10 and v = -12 are indeed solutions. This comprehensive approach not only solves the specific problem but also provides a framework for tackling similar inequality problems. The ability to solve inequalities is crucial for higher-level mathematics and various real-world applications, making it a valuable skill to master.

Throughout this article, we've explored the process of solving the inequality 47 < -5-8v, emphasizing the importance of each step. We began by understanding the inequality, then systematically isolated the variable 'v', remembering to reverse the inequality sign when dividing by a negative number. This led us to the solution v < -6.5. Next, we verified our solution by substituting the given values into the original inequality. This step is critical to ensure the accuracy of our solution and to identify any potential errors. The results confirmed that v = -10 and v = -12 are solutions, while v = 7 and v = 2 are not. This comprehensive approach, combining algebraic manipulation with verification, is the key to mastering inequality solutions. By following these steps, you can confidently tackle a wide range of inequality problems.