Evaluating Inequalities Which Are True When X Equals -10

This article provides a detailed walkthrough of how to determine which inequalities hold true when $x = -10$. We'll evaluate each inequality step-by-step, ensuring a clear understanding of the process. This topic falls under the crucial realm of mathematical inequalities, a fundamental concept in algebra and calculus. Understanding how to substitute values into inequalities and determine their validity is essential for problem-solving in various mathematical contexts.

Understanding Inequalities

Before diving into the specific problem, let's establish a firm understanding of inequalities. Inequalities are mathematical statements that compare two expressions using symbols like greater than ($>$), less than ($ extless$), greater than or equal to ($ extgreater=$), and less than or equal to ($ extless=$). Unlike equations, which assert the equality of two expressions, inequalities indicate a range of possible values that satisfy the relationship. When solving inequalities, we often manipulate them algebraically, much like equations, but with a crucial difference: multiplying or dividing by a negative number reverses the direction of the inequality sign. This rule is paramount in ensuring accurate solutions.

In this particular problem, we are given a set of inequalities involving the variable $x$, and our task is to determine which of these inequalities hold true when $x$ is specifically equal to $-10$. This involves substituting $-10$ for $x$ in each inequality and then simplifying the resulting numerical expressions to see if the inequality relationship holds. For instance, if we have the inequality $x + 5 > 0$, substituting $x = -10$ gives us $-10 + 5 > 0$, which simplifies to $-5 > 0$. This statement is false, so the inequality does not hold true for $x = -10$. This example illustrates the basic process we will use to evaluate each of the given inequalities.

By mastering the skill of evaluating inequalities for specific values, you build a solid foundation for solving more complex problems involving inequalities, such as those found in calculus, optimization, and mathematical modeling. The ability to correctly manipulate and interpret inequalities is not just a mathematical skill but also a valuable tool for logical reasoning and problem-solving in various real-world scenarios. Therefore, a thorough understanding of this topic is crucial for anyone pursuing further studies in mathematics or related fields.

The Inequalities to Evaluate

We are presented with the following inequalities:

1. $9x - 5 > 84.5$
2. $-9x - 5 < 84.5$
3. $-9x - 5 > -84.5$
4. $-9x - 5 < -84.5$

Our objective is to substitute $x = -10$ into each of these inequalities and determine which ones hold true. Each inequality presents a slightly different scenario, allowing us to explore various aspects of inequality evaluation. The first inequality, $9x - 5 > 84.5$, involves a positive coefficient for $x$, whereas the remaining inequalities involve a negative coefficient, which, as we discussed earlier, is a crucial point to consider when manipulating inequalities. Understanding how the sign of the coefficient affects the inequality is essential for accurate evaluation.

The inequalities also differ in the constant term on the right-hand side, which varies between positive and negative values. This variation allows us to observe how the magnitude and sign of the constant term influence the truth of the inequality. By analyzing these inequalities, we reinforce our understanding of how different numerical values interact within the context of inequalities.

To effectively evaluate these inequalities, we will follow a systematic approach: First, we substitute $x = -10$ into each inequality. Then, we simplify the resulting expression on the left-hand side by performing the necessary arithmetic operations. Finally, we compare the simplified value with the constant term on the right-hand side to determine if the inequality holds true. This step-by-step process not only ensures accuracy but also promotes a clear and organized approach to problem-solving. By carefully examining each step, we can develop a deeper understanding of the underlying concepts and improve our ability to tackle similar problems in the future.

Evaluating Inequality 1: $9x - 5 > 84.5$

Let's start by evaluating the first inequality: $9x - 5 > 84.5$. To do this, we substitute $x = -10$ into the inequality:

$9(-10) - 5 > 84.5$

Now, we simplify the left side of the inequality. First, we multiply $9$ by $-10$, which gives us $-90$:

$-90 - 5 > 84.5$

Next, we subtract $5$ from $-90$:

$-95 > 84.5$

The resulting statement is $-95 > 84.5$. We need to determine if this inequality is true or false. Clearly, $-95$ is a negative number, and $84.5$ is a positive number. Negative numbers are always less than positive numbers. Therefore, $-95$ is not greater than $84.5$.

Thus, the inequality $-95 > 84.5$ is false. This means that the original inequality, $9x - 5 > 84.5$, does not hold true when $x = -10$. This step-by-step evaluation demonstrates the importance of carefully performing each arithmetic operation and then accurately comparing the resulting values to determine the validity of the inequality. By breaking down the problem into smaller, manageable steps, we can minimize the risk of errors and ensure a clear understanding of the solution process. This approach is particularly valuable when dealing with more complex inequalities or systems of inequalities.

Evaluating Inequality 2: $-9x - 5 < 84.5$

Now, let's evaluate the second inequality: $-9x - 5 < 84.5$. We substitute $x = -10$ into the inequality:

$-9(-10) - 5 < 84.5$

First, we multiply $-9$ by $-10$, which gives us $90$:

$90 - 5 < 84.5$

Next, we subtract $5$ from $90$:

$85 < 84.5$

The resulting statement is $85 < 84.5$. We need to determine if this inequality is true or false. Here, $85$ is greater than $84.5$, so the statement $85 < 84.5$ is false. This indicates that the original inequality, $-9x - 5 < 84.5$, does not hold true when $x = -10$.

This evaluation highlights the importance of paying close attention to the order of operations and the signs of the numbers involved. Multiplying two negative numbers results in a positive number, which can significantly affect the outcome of the inequality. By carefully tracking each step and comparing the final values, we can confidently determine the validity of the inequality. This methodical approach is crucial for avoiding common errors and ensuring accurate solutions, especially when dealing with inequalities involving negative coefficients or multiple operations.

Evaluating Inequality 3: $-9x - 5 > -84.5$

Let's proceed to the third inequality: $-9x - 5 > -84.5$. We substitute $x = -10$ into the inequality:

$-9(-10) - 5 > -84.5$

First, we multiply $-9$ by $-10$, which gives us $90$:

$90 - 5 > -84.5$

Next, we subtract $5$ from $90$:

$85 > -84.5$

The resulting statement is $85 > -84.5$. We need to determine if this inequality is true or false. Since $85$ is a positive number and $-84.5$ is a negative number, $85$ is indeed greater than $-84.5$. Therefore, the inequality $85 > -84.5$ is true. This means that the original inequality, $-9x - 5 > -84.5$, holds true when $x = -10$.

This evaluation further reinforces the concept that positive numbers are always greater than negative numbers. By carefully simplifying the inequality and comparing the resulting values, we can confidently determine its validity. This skill is essential for solving inequalities and understanding the relationships between numbers on the number line. Moreover, this example demonstrates how inequalities can be used to describe a range of values that satisfy a given condition, a fundamental concept in mathematics and its applications.

Evaluating Inequality 4: $-9x - 5 < -84.5$

Finally, let's evaluate the fourth inequality: $-9x - 5 < -84.5$. We substitute $x = -10$ into the inequality:

$-9(-10) - 5 < -84.5$

First, we multiply $-9$ by $-10$, which gives us $90$:

$90 - 5 < -84.5$

Next, we subtract $5$ from $90$:

$85 < -84.5$

The resulting statement is $85 < -84.5$. We need to determine if this inequality is true or false. Since $85$ is a positive number and $-84.5$ is a negative number, $85$ is not less than $-84.5$. Therefore, the inequality $85 < -84.5$ is false. This means that the original inequality, $-9x - 5 < -84.5$, does not hold true when $x = -10$.

This final evaluation underscores the importance of accurately comparing positive and negative numbers. A positive number will never be less than a negative number, so the inequality $85 < -84.5$ is clearly false. This example reinforces the fundamental concepts of inequalities and their relationships to the number line. By systematically evaluating each inequality, we have developed a comprehensive understanding of how to determine the validity of inequalities for specific values of the variable.

Conclusion

In conclusion, after evaluating each inequality at $x = -10$, we found that only the inequality $-9x - 5 > -84.5$ holds true. This exercise demonstrates the process of substituting values into inequalities and simplifying to determine their validity. Understanding how to work with inequalities is a crucial skill in mathematics, with applications in various fields, including algebra, calculus, and optimization. By mastering these fundamental concepts, you are well-prepared to tackle more complex mathematical problems and real-world applications.

The key takeaways from this analysis include:

• Substituting a given value for a variable in an inequality.
• Following the order of operations to simplify the resulting expression.
• Carefully comparing the simplified value with the constant term to determine the truth of the inequality.
• Understanding the relationship between positive and negative numbers in the context of inequalities.

By applying these principles, you can confidently evaluate inequalities and solve a wide range of mathematical problems. Remember to always pay close attention to the details, such as the signs of the numbers and the direction of the inequality, to ensure accurate results. With practice and a solid understanding of the fundamentals, you can excel in this area of mathematics.