Solving -13x > -39 A Step By Step Guide

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In mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Unlike equations, which assert the equality of two expressions, inequalities express relationships such as "greater than," "less than," "greater than or equal to," or "less than or equal to." This article delves into the process of solving a specific inequality: -13x > -39. We will explore the steps involved, the underlying principles, and why certain operations must be performed with care when dealing with inequalities.

The inequality -13x > -39 presents a scenario where the product of -13 and an unknown value 'x' is greater than -39. Our goal is to isolate 'x' on one side of the inequality to determine the range of values that satisfy this condition. This process involves algebraic manipulations similar to solving equations, but with a crucial difference: multiplying or dividing by a negative number requires a change in the direction of the inequality sign. Understanding this rule is paramount to correctly solving inequalities.

Before diving into the solution, it's essential to grasp the concept of inequality signs and their implications. The "greater than" symbol (>) indicates that the value on the left is larger than the value on the right. Conversely, the "less than" symbol (<) signifies that the value on the left is smaller than the value on the right. When we multiply or divide by a negative number, we are essentially reflecting the number line, which necessitates flipping the inequality sign to maintain the truth of the statement. The article will guide you through the step-by-step solution of the inequality -13x > -39, ensuring a clear understanding of each step and the reasoning behind it. We will also discuss common pitfalls and how to avoid them, making this a comprehensive guide to solving similar problems in the future.

To solve the inequality -13x > -39, our primary objective is to isolate the variable 'x' on one side. This involves performing algebraic operations on both sides of the inequality while maintaining its balance. The key operation in this case is division, but we must be mindful of the sign of the number we are dividing by. Since we are dividing by a negative number (-13), we must remember to flip the inequality sign to ensure the solution remains accurate.

The initial inequality is -13x > -39. To isolate 'x', we need to divide both sides of the inequality by -13. This gives us: (-13x) / (-13) and (-39) / (-13). On the left side, -13x divided by -13 simplifies to 'x'. On the right side, -39 divided by -13 equals 3. However, because we divided by a negative number, we must flip the "greater than" sign (>) to a "less than" sign (<). This crucial step is often the source of errors in solving inequalities, so it's important to emphasize and understand its significance.

After performing the division and flipping the inequality sign, we arrive at the solution: x < 3. This means that any value of 'x' that is less than 3 will satisfy the original inequality -13x > -39. To verify this solution, we can substitute a value less than 3 into the original inequality and check if the inequality holds true. For example, let's substitute x = 0: -13(0) > -39, which simplifies to 0 > -39. This is a true statement, confirming that our solution x < 3 is correct. Similarly, we can test a value greater than or equal to 3 to see that it does not satisfy the original inequality. For instance, if x = 3, then -13(3) > -39 becomes -39 > -39, which is false. This reinforces the understanding that the solution set consists only of values strictly less than 3. This detailed step-by-step explanation should provide a clear understanding of how to solve the inequality and the importance of flipping the inequality sign when dividing by a negative number.

The critical aspect of solving the inequality -13x > -39 lies in understanding the rule of flipping the inequality sign when dividing (or multiplying) by a negative number. This rule is not just a mathematical quirk; it stems from the fundamental properties of inequalities and the number line. To fully grasp why this rule exists, let's delve into the underlying principles.

Inequalities express the relative order of two values. For instance, the statement 5 > 2 signifies that 5 is greater than 2. When we multiply or divide both sides of an inequality by a positive number, we are essentially scaling the number line, but the relative order of the numbers remains the same. For example, if we multiply both sides of 5 > 2 by 3, we get 15 > 6, which is still a true statement. However, when we multiply or divide by a negative number, we are not only scaling the number line but also reflecting it across the origin. This reflection reverses the order of the numbers.

Consider the inequality 2 < 4. If we multiply both sides by -1, we get -2 and -4. On the number line, -2 is to the right of -4, meaning -2 is greater than -4. Therefore, multiplying by -1 changes the relationship from 2 < 4 to -2 > -4. This demonstrates why the inequality sign must be flipped to maintain the truth of the statement. Applying this principle to the inequality -13x > -39, dividing both sides by -13 not only scales the values but also reflects the number line, necessitating the change from “greater than” to “less than.” Without flipping the sign, the solution would be incorrect, leading to a misunderstanding of the values that satisfy the inequality. The rule of flipping the inequality sign is therefore not arbitrary but a direct consequence of how negative numbers affect the order of values on the number line.

After carefully solving the inequality -13x > -39, we arrived at the solution x < 3. This solution indicates that any value of 'x' that is strictly less than 3 will satisfy the original inequality. Now, let's examine the provided options and determine the correct answer along with the reasons why the other options are incorrect.

The options presented were:

A. x > -3 B. x < 3 C. x < -3 D. x > 3

Based on our step-by-step solution, the correct answer is B. x < 3. This option accurately represents the range of values for 'x' that make the inequality -13x > -39 true. As we demonstrated earlier, dividing both sides of the inequality by -13 and flipping the inequality sign yields x < 3.

Now, let's analyze why the other options are incorrect:

  • A. x > -3: This option suggests that 'x' must be greater than -3. If we substitute a value like x = 0 (which is greater than -3) into the original inequality, we get -13(0) > -39, which simplifies to 0 > -39. This is true. However, if we substitute x = 4 (which is also greater than -3), we get -13(4) > -39, which simplifies to -52 > -39. This is false. Therefore, x > -3 is not the correct solution because it includes values that do not satisfy the inequality.
  • C. x < -3: This option implies that 'x' must be less than -3. If we substitute a value like x = -4 (which is less than -3) into the original inequality, we get -13(-4) > -39, which simplifies to 52 > -39. This is true. However, this option excludes values between -3 and 3, which, as we know from the correct solution, should be included. For example, if x = 0, the inequality holds true, but this value is not included in x < -3.
  • D. x > 3: This option suggests that 'x' must be greater than 3. If we substitute a value like x = 4 (which is greater than 3) into the original inequality, we get -13(4) > -39, which simplifies to -52 > -39. This is false. Therefore, x > 3 is not a valid solution.

In conclusion, option B, x < 3, is the only correct answer because it accurately represents the solution set of the inequality -13x > -39. The other options either include values that do not satisfy the inequality or exclude values that should be included.

When solving inequalities, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls and understanding how to avoid them is crucial for mastering inequality problems. This section will highlight some of the most frequent errors and provide strategies to ensure accuracy.

  • Forgetting to Flip the Inequality Sign: The most common mistake is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. As discussed earlier, this step is essential because multiplying or dividing by a negative number reverses the order of the numbers on the number line. To avoid this mistake, always double-check the sign of the number you are multiplying or dividing by. If it's negative, make a conscious effort to flip the inequality sign. A helpful technique is to circle the negative number as a reminder to perform the sign flip.

  • Incorrectly Applying the Distributive Property: Another common error occurs when dealing with inequalities that involve parentheses. The distributive property must be applied correctly to remove the parentheses before proceeding with other operations. Ensure that each term inside the parentheses is multiplied by the term outside. For example, in the inequality -2(x + 3) < 4, the -2 must be multiplied by both 'x' and '3', resulting in -2x - 6 < 4. A mistake in this step can lead to an entirely incorrect solution.

  • Combining Like Terms Incorrectly: Similar to solving equations, combining like terms is a fundamental step in solving inequalities. Ensure that you only combine terms that have the same variable and exponent. For example, in the inequality 3x + 2 - x > 5, the terms 3x and -x can be combined to give 2x + 2 > 5. Carelessly combining unlike terms will result in an inaccurate inequality.

  • Incorrectly Interpreting the Solution Set: After solving an inequality, it's essential to correctly interpret the solution set. For example, x < 3 means that 'x' can be any value strictly less than 3, but not equal to 3. Representing the solution set on a number line can be helpful in visualizing the range of values. Use an open circle for strict inequalities (< or >) and a closed circle for inclusive inequalities (≤ or ≥).

  • Not Verifying the Solution: A simple yet effective way to avoid mistakes is to verify the solution by substituting a value from the solution set back into the original inequality. If the inequality holds true, it provides confidence in the correctness of the solution. Additionally, substituting a value outside the solution set should result in a false statement, further confirming the solution.

By being mindful of these common mistakes and implementing strategies to avoid them, you can significantly improve your accuracy in solving inequalities.

In conclusion, solving inequalities is a fundamental skill in mathematics that requires a solid understanding of algebraic principles and careful attention to detail. This article has provided a comprehensive guide to solving the inequality -13x > -39, emphasizing the critical step of flipping the inequality sign when dividing by a negative number. We have explored the step-by-step solution, discussed the reasoning behind the rule, identified the correct answer, and explained why other options are incorrect. Furthermore, we have highlighted common mistakes and provided strategies to avoid them, ensuring a thorough understanding of the topic.

The key takeaway from this discussion is the importance of precision and accuracy when working with inequalities. The simple act of dividing by a negative number can significantly alter the relationship between values, necessitating a change in the inequality sign. Mastering this rule, along with other algebraic techniques, is essential for solving a wide range of inequality problems.

By understanding the concepts presented in this article, you can confidently tackle similar inequalities and apply these principles to more complex mathematical problems. Remember to always double-check your work, verify your solutions, and pay close attention to the signs and operations involved. With practice and a clear understanding of the rules, you can master inequalities and enhance your mathematical skills.

The ability to solve inequalities is not just limited to academic exercises; it has practical applications in various fields, including economics, engineering, and computer science. Understanding inequalities allows us to model and solve real-world problems involving constraints and limitations. Therefore, mastering this skill is a valuable asset that will benefit you in various aspects of your academic and professional life. Keep practicing, keep exploring, and keep mastering the world of mathematics!