Solving -(x/3) ≥ 5 A Step-by-Step Guide

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This article provides a comprehensive walkthrough on how to solve the linear inequality -(x/3) ≥ 5. Inequalities are mathematical statements that compare two expressions using symbols like 'greater than' (>) , 'less than' (<), 'greater than or equal to' (≥), and 'less than or equal to' (≤). Solving inequalities involves finding the range of values that satisfy the given condition. This guide will meticulously explain each step, ensuring a clear understanding of the process. Our focus will be on isolating the variable x while adhering to the rules that govern inequality manipulation, particularly the crucial step of handling negative coefficients. By the end of this guide, you'll confidently tackle similar problems and grasp the underlying principles of inequality solutions.

Understanding the Basics of Inequalities

Before diving into the solution, let's solidify our understanding of inequalities. Unlike equations that have definite solutions, inequalities represent a range of possible values. An inequality uses symbols to express relationships other than equality. For example, a > b means 'a' is greater than 'b', and a ≤ b means 'a' is less than or equal to 'b'. When solving inequalities, the goal remains the same as with equations: to isolate the variable. However, there's a critical rule to remember: multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. This is because multiplying or dividing by a negative number changes the direction of the number line, effectively flipping the order of the numbers. This concept is crucial for accurately solving inequalities like the one we're addressing, where the variable x is multiplied by a negative coefficient. Failing to account for this rule leads to incorrect solutions, emphasizing the need for careful application of algebraic principles. Understanding these basics sets the foundation for confidently manipulating and solving a wide array of inequality problems.

Step-by-Step Solution of -(x/3) ≥ 5

To solve the inequality -(x/3) ≥ 5, we will follow a series of algebraic steps to isolate the variable x. The primary goal is to get x by itself on one side of the inequality, revealing the range of values that satisfy the condition. Let's break down the process:

Step 1: Multiply Both Sides by -3

The first step involves eliminating the fraction and the negative sign associated with the x term. To do this, we multiply both sides of the inequality by -3. Remember the crucial rule: multiplying or dividing an inequality by a negative number reverses the inequality sign. So, when we multiply by -3, the '≥' sign will flip to '≤'.

  • Original inequality: -(x/3) ≥ 5
  • Multiply both sides by -3: (-3) * -(x/3) ≤ (-3) * 5

Step 2: Simplify the Inequality

Now, we simplify both sides of the inequality.

  • On the left side, the -3 and the /3 cancel out, and the two negative signs multiply to a positive sign, leaving us with x.
  • On the right side, we multiply -3 by 5, which results in -15.
  • Simplified inequality: x ≤ -15

Step 3: Interpret the Solution

The solution x ≤ -15 means that x can be any value less than or equal to -15. This represents a range of values, extending from negative infinity up to and including -15. To fully understand this solution, consider visualizing it on a number line. A closed circle or a filled-in dot is placed on -15 to indicate that -15 is included in the solution set, and an arrow extends to the left, representing all values less than -15. This graphical representation helps solidify the concept that inequalities provide a set of solutions rather than a single value, as is the case with equations.

Detailed Explanation of Each Step

Let's delve deeper into each step to ensure a thorough understanding of the solution process for the inequality -(x/3) ≥ 5.

Multiplying by a Negative Number and Reversing the Inequality Sign

This is the most critical aspect of solving this inequality. When we multiply both sides of -(x/3) ≥ 5 by -3, we're not just changing the numbers; we're changing the direction of the inequality. This stems from how negative numbers behave on the number line. Multiplying by a negative number reflects the numbers across zero, effectively swapping their positions relative to each other. Imagine two numbers, say 2 and 5, where 2 < 5. If we multiply both by -1, we get -2 and -5. Now, -2 is greater than -5 (-2 > -5). This illustrates why the inequality sign must be flipped. In our case, multiplying by -3 ensures that the negative sign associated with x is eliminated, allowing us to isolate x. Forgetting to reverse the inequality sign is a common error that leads to an incorrect solution set, highlighting the importance of mastering this fundamental rule.

Simplifying the Inequality and Isolating x

After multiplying both sides by -3, the inequality becomes (-3) * -(x/3) ≤ (-3) * 5. The left side simplifies because the -3 in the numerator cancels out the 3 in the denominator, and the two negative signs become positive. This leaves us with just x on the left side. On the right side, -3 multiplied by 5 gives us -15. This simplification results in the inequality x ≤ -15. The variable x is now isolated, meaning we've successfully determined the condition it must satisfy. This step is crucial for clearly defining the solution set and understanding the range of values that fulfill the original inequality. The simplification process showcases the power of algebraic manipulation in making complex expressions more manageable and revealing the underlying relationships between variables.

Interpreting the Solution Set x ≤ -15

The solution x ≤ -15 is not a single value but a set of values. It includes -15 and every number less than -15. To fully grasp this, visualize a number line. Place a closed circle (or a filled-in dot) on -15, indicating that -15 is part of the solution. Then, draw an arrow extending to the left from -15, signifying that all numbers to the left of -15 are also solutions. This visual representation underscores that x can take on an infinite number of values as long as they are less than or equal to -15. Examples of solutions include -15, -16, -15.5, -20, and so on. Understanding the solution set in this way helps prevent the misconception that inequalities have only one answer, as is typical with equations. Instead, inequalities define a range or interval of values that satisfy the given condition, making the solution set a fundamental concept in inequality problems.

Common Mistakes to Avoid

When solving inequalities, several common mistakes can lead to incorrect answers. Recognizing and avoiding these pitfalls is crucial for accuracy. Let's discuss the most frequent errors:

Forgetting to Reverse the Inequality Sign

The most critical mistake, as previously emphasized, is failing to flip the inequality sign when multiplying or dividing both sides by a negative number. This error stems from treating inequalities like equations, where the sign remains unchanged regardless of the operation. To avoid this, always double-check whether you're multiplying or dividing by a negative value and make the necessary adjustment to the inequality sign. In our example, when multiplying -(x/3) ≥ 5 by -3, the '≥' must change to '≤'.

Incorrectly Distributing Negative Signs

Another common error occurs when distributing negative signs, especially if the inequality involves expressions within parentheses or fractions. Ensure that the negative sign is correctly applied to every term inside the parentheses or to the entire numerator of a fraction. For instance, if the inequality were -2(x + 3) > 4, you would need to distribute the -2 to both x and 3, resulting in -2x - 6 > 4. An incorrect distribution can alter the entire problem and lead to a wrong solution.

Misinterpreting the Solution Set

Inequalities represent a range of values, not a single solution. Misinterpreting the solution set can lead to incorrect conclusions. For example, if the solution is x < 5, it includes all numbers less than 5 but not 5 itself. It's important to differentiate between strict inequalities (>, <) and inclusive inequalities (≥, ≤). Strict inequalities do not include the endpoint, while inclusive inequalities do. Visualizing the solution on a number line can help avoid this error. Use an open circle for strict inequalities and a closed circle for inclusive inequalities.

Performing Operations in the Wrong Order

Just like with equations, the order of operations (PEMDAS/BODMAS) is crucial when solving inequalities. Ensure that you perform operations in the correct sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Ignoring the order of operations can lead to incorrect simplifications and, consequently, an incorrect solution.

Practice Problems

To solidify your understanding of solving inequalities, let's work through a few practice problems. These examples will cover different scenarios and help you apply the concepts we've discussed.

Problem 1: Solve 2x + 3 < 7

Solution:

  1. Subtract 3 from both sides: 2x < 4
  2. Divide both sides by 2: x < 2
  3. The solution set includes all numbers less than 2.

Problem 2: Solve -4x - 5 ≥ 3

Solution:

  1. Add 5 to both sides: -4x ≥ 8
  2. Divide both sides by -4 (and reverse the inequality sign): x ≤ -2
  3. The solution set includes all numbers less than or equal to -2.

Problem 3: Solve 3(x - 2) > 6

Solution:

  1. Distribute the 3: 3x - 6 > 6
  2. Add 6 to both sides: 3x > 12
  3. Divide both sides by 3: x > 4
  4. The solution set includes all numbers greater than 4.

Problem 4: Solve -(x/2) + 1 ≤ 4

Solution:

  1. Subtract 1 from both sides: -(x/2) ≤ 3
  2. Multiply both sides by -2 (and reverse the inequality sign): x ≥ -6
  3. The solution set includes all numbers greater than or equal to -6.

By working through these practice problems, you can reinforce your ability to solve inequalities accurately and confidently. Remember to pay close attention to the rules of manipulating inequalities, especially when dealing with negative numbers.

Conclusion

In conclusion, solving inequalities is a fundamental skill in mathematics with applications across various fields. Throughout this guide, we've meticulously dissected the process of solving the inequality -(x/3) ≥ 5, emphasizing each step to ensure clarity and comprehension. We've explored the crucial rule of reversing the inequality sign when multiplying or dividing by a negative number, a cornerstone of accurate inequality manipulation. Furthermore, we've highlighted common mistakes to avoid, such as incorrect distribution of negative signs and misinterpretation of solution sets, empowering you to tackle inequality problems with increased confidence. By understanding that inequalities represent a range of solutions rather than a single value, and by visualizing solution sets on a number line, you can develop a deeper understanding of the concepts. The practice problems provided offer an opportunity to apply these newly acquired skills and solidify your proficiency in solving inequalities. Mastering these concepts not only enhances your mathematical abilities but also provides a valuable toolkit for problem-solving in diverse real-world scenarios. Remember, consistent practice and a keen awareness of the underlying principles are key to achieving mastery in solving inequalities and other mathematical challenges. We hope this comprehensive guide has equipped you with the knowledge and skills necessary to confidently solve linear inequalities and excel in your mathematical endeavors. Keep practicing, and you'll find that solving inequalities becomes second nature.

Therefore, the correct answer is B. x ≤ -15