Solving Inequalities A Step By Step Guide For -7x + 9 > -8

Jun 27, 2025 by ADMIN 59 views

Solve the inequality -7x + 9 > -8. Which of the following is the correct solution? A. x > 17/7 B. x < 17/7 C. x ≥ -1/7 D. x < -1/7

Solving Inequalities: A Step-by-Step Guide to -7x + 9 > -8

In the realm of mathematics, inequalities play a crucial role in defining relationships between quantities that are not necessarily equal. Unlike equations, which establish equality, inequalities express a range of possible values. This article delves into the process of solving a specific linear inequality: -7x + 9 > -8. We will break down each step, providing a comprehensive understanding of the underlying principles and techniques. By the end of this exploration, you'll be well-equipped to tackle similar inequality problems with confidence.

Understanding the Fundamentals of Inequalities

Before we dive into the solution, let's solidify our understanding of the basic concepts. Inequalities are mathematical statements that compare two expressions using symbols like '>', '<', '≥', and '≤'. These symbols represent 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to', respectively. Solving an inequality involves isolating the variable on one side of the inequality symbol, just like solving an equation. However, a crucial difference arises when multiplying or dividing both sides by a negative number: we must flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the number line.

For instance, if we have 2 < 4, multiplying both sides by -1 gives -2 > -4. This sign reversal is essential to maintain the truth of the inequality. With these foundational principles in mind, let's embark on solving our given inequality: -7x + 9 > -8.

Step-by-Step Solution of -7x + 9 > -8

1. Isolating the Term with the Variable

The first step in solving any inequality is to isolate the term containing the variable. In our case, this means isolating -7x. To do this, we need to eliminate the +9 on the left side of the inequality. We can achieve this by subtracting 9 from both sides of the inequality. Remember, whatever operation we perform on one side, we must perform on the other to maintain the balance of the inequality.

-7x + 9 > -8

Subtracting 9 from both sides, we get:

-7x + 9 - 9 > -8 - 9

This simplifies to:

-7x > -17

We have successfully isolated the term with the variable, -7x. Now, we move on to the next step, which involves isolating the variable itself.

2. Isolating the Variable

Our goal now is to isolate 'x'. Currently, 'x' is being multiplied by -7. To isolate 'x', we need to divide both sides of the inequality by -7. This is where the crucial rule about flipping the inequality sign comes into play. Since we are dividing by a negative number, we must reverse the inequality sign to maintain the truth of the statement.

-7x > -17

Dividing both sides by -7, we get:

(-7x) / -7 < (-17) / -7

Notice that the '>' sign has been flipped to '<'. This simplifies to:

x < 17/7

3. Expressing the Solution

We have now arrived at the solution: x < 17/7. This inequality states that 'x' can be any value less than 17/7. It's crucial to understand that this represents an infinite range of values. To visualize this solution, we can represent it on a number line. Draw a number line and mark the point 17/7. Since the inequality is 'less than' (and not 'less than or equal to'), we use an open circle at 17/7 to indicate that this value is not included in the solution. Then, we shade the region to the left of 17/7, representing all the values that satisfy the inequality.

Alternatively, we can express the solution in interval notation. Interval notation uses parentheses and brackets to represent ranges of values. Parentheses indicate that the endpoint is not included, while brackets indicate that it is included. For our solution, x < 17/7, the interval notation is (-∞, 17/7). The parenthesis next to -∞ indicates that negative infinity is not a specific number and is never included in an interval. The parenthesis next to 17/7 indicates that 17/7 is not included in the solution set.

Verifying the Solution

To ensure the correctness of our solution, it's always a good practice to verify it. We can do this by picking a value within our solution range (x < 17/7) and substituting it back into the original inequality. If the inequality holds true, our solution is likely correct. Let's choose x = 2 as a test value, since 2 is less than 17/7 (which is approximately 2.43).

Substituting x = 2 into the original inequality -7x + 9 > -8, we get:

-7(2) + 9 > -8

-14 + 9 > -8

-5 > -8

This statement is true, as -5 is indeed greater than -8. This confirms that our solution, x < 17/7, is correct. If we had obtained a false statement, it would indicate an error in our solution process, and we would need to revisit our steps.

Analyzing Potential Pitfalls and Common Mistakes

Solving inequalities, while conceptually similar to solving equations, presents some unique challenges. One common mistake is forgetting to flip the inequality sign when multiplying or dividing by a negative number. This can lead to an incorrect solution. Another pitfall is misinterpreting the inequality symbols. It's crucial to remember the difference between '>', '<', '≥', and '≤' and to use them correctly in expressing the solution.

Additionally, when dealing with more complex inequalities involving multiple steps, it's easy to make arithmetic errors. Therefore, it's essential to perform each step carefully and double-check your calculations. Regular practice and a thorough understanding of the underlying principles are key to avoiding these pitfalls and mastering the art of solving inequalities.

Conclusion: Mastering the Art of Solving Inequalities

In this comprehensive guide, we have dissected the process of solving the linear inequality -7x + 9 > -8. We began by establishing the fundamental principles of inequalities, emphasizing the crucial rule of flipping the inequality sign when multiplying or dividing by a negative number. We then meticulously walked through each step of the solution, isolating the variable and expressing the solution in both inequality and interval notation.

Furthermore, we highlighted the importance of verifying the solution to ensure its accuracy and discussed common pitfalls to avoid. By understanding these concepts and practicing regularly, you can develop a strong foundation in solving inequalities and confidently tackle a wide range of mathematical problems. The ability to solve inequalities is not just a valuable skill in mathematics but also a crucial tool in various fields, including science, engineering, and economics, where understanding relationships between quantities is paramount.

Are you grappling with the inequality -7x + 9 > -8? Do you want to understand the step-by-step process of solving it? This guide provides a comprehensive solution, ensuring you grasp the underlying mathematical principles. We will break down each step, explain the reasoning, and highlight potential pitfalls to avoid. By the end of this article, you'll be equipped to solve similar inequalities with confidence.

Understanding the Basics of Inequalities

Before diving into the solution, let's revisit the fundamental concepts of inequalities. Unlike equations, which express equality between two expressions, inequalities express a relationship where two expressions are not necessarily equal. The key symbols used in inequalities are:

> (greater than)
< (less than)
≥ (greater than or equal to)
≤ (less than or equal to)

The goal of solving an inequality is to isolate the variable (in this case, 'x') on one side of the inequality symbol. The solution will represent a range of values that satisfy the inequality, rather than a single value as in an equation.

A critical rule to remember is that when you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the number line. For example, if 2 < 4, then multiplying both sides by -1 gives -2 > -4.

Step-by-Step Solution of -7x + 9 > -8

Let's now tackle the inequality -7x + 9 > -8 step by step.

1. Isolate the Term with 'x'

The first step is to isolate the term containing the variable 'x', which is -7x. To do this, we need to eliminate the +9 on the left side of the inequality. We achieve this by subtracting 9 from both sides:

-7x + 9 - 9 > -8 - 9

This simplifies to:

-7x > -17

2. Isolate 'x'

Now, we need to isolate 'x' itself. Currently, 'x' is being multiplied by -7. To isolate 'x', we divide both sides of the inequality by -7. Remember the crucial rule: since we are dividing by a negative number, we must flip the inequality sign:

(-7x) / -7 < (-17) / -7

This simplifies to:

x < 17/7

Therefore, the solution to the inequality is x < 17/7.

3. Expressing the Solution Set

The solution x < 17/7 means that any value of 'x' less than 17/7 will satisfy the original inequality. We can represent this solution in a few ways:

Inequality Notation: x < 17/7
Interval Notation: (-∞, 17/7)
Number Line: A number line with an open circle at 17/7 and shading to the left, indicating all values less than 17/7.

4. Verifying the Solution

It's always a good practice to verify your solution. To do this, choose a value of 'x' that falls within your solution set (x < 17/7) and substitute it back into the original inequality. If the inequality holds true, your solution is likely correct.

Let's choose x = 2, which is less than 17/7 (approximately 2.43).

Substitute x = 2 into the original inequality:

-7(2) + 9 > -8

-14 + 9 > -8

-5 > -8

This statement is true, so our solution x < 17/7 is correct.

Common Mistakes and How to Avoid Them

Solving inequalities can be tricky, and certain mistakes are common. Here are some to watch out for:

Forgetting to Flip the Inequality Sign: This is the most common mistake. Always remember to flip the inequality sign when multiplying or dividing both sides by a negative number.
Arithmetic Errors: Double-check your calculations at each step to avoid simple arithmetic errors.
Misinterpreting Inequality Symbols: Ensure you understand the difference between >, <, ≥, and ≤.
Incorrectly Expressing the Solution Set: Be careful when using interval notation and number lines to represent the solution set. Make sure you use parentheses for open intervals (not including the endpoint) and brackets for closed intervals (including the endpoint).

Practice Problems

To solidify your understanding, try solving these practice problems:

-3x + 5 < 14
2x - 7 ≥ -1
-5x + 10 > 0

Conclusion: Mastering Inequalities

Solving inequalities is a fundamental skill in mathematics with applications in various fields. By understanding the basic principles, following the steps carefully, and avoiding common mistakes, you can master the art of solving inequalities. This guide has provided you with a comprehensive approach to solving -7x + 9 > -8 and similar problems. Remember to practice regularly to build your confidence and proficiency.

In the realm of algebra, linear inequalities serve as powerful tools for expressing relationships where quantities are not necessarily equal. Unlike equations that pinpoint specific values, inequalities define ranges or intervals within which solutions lie. This comprehensive guide delves into the intricacies of solving the linear inequality -7x + 9 > -8, offering a step-by-step approach coupled with insightful explanations and practical tips. Whether you're a student seeking to enhance your understanding or an educator looking for a clear teaching resource, this article aims to equip you with the knowledge and confidence to tackle linear inequalities effectively.

Understanding the Essence of Linear Inequalities

Before embarking on the solution process, it's crucial to grasp the core concepts underpinning linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as >, <, ≥, and ≤, which represent 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to', respectively. Solving a linear inequality involves isolating the variable on one side of the inequality symbol, mirroring the process of solving linear equations. However, a pivotal distinction arises when multiplying or dividing both sides by a negative number: the direction of the inequality sign must be reversed. This reversal is essential to preserve the validity of the inequality, as multiplying or dividing by a negative value effectively flips the number line.

For instance, consider the inequality 2 < 4. If we multiply both sides by -1, we obtain -2 > -4. The inequality sign flips from '<' to '>' to accurately reflect the relationship between -2 and -4. With these fundamental principles in mind, let's proceed to solve the inequality -7x + 9 > -8.

A Step-by-Step Journey to Solving -7x + 9 > -8

1. Isolating the Variable Term

The initial step in solving any inequality is to isolate the term containing the variable. In our case, this entails isolating -7x. To achieve this, we must eliminate the +9 term on the left side of the inequality. This can be accomplished by subtracting 9 from both sides. Remember, whatever operation we perform on one side, we must replicate it on the other to maintain the equilibrium of the inequality.

-7x + 9 > -8

Subtracting 9 from both sides yields:

-7x + 9 - 9 > -8 - 9

This simplifies to:

-7x > -17

We have successfully isolated the variable term, -7x. Now, we transition to the next step, which involves isolating the variable itself.

2. Isolating the Variable 'x'

Our current objective is to isolate 'x'. Currently, 'x' is being multiplied by -7. To isolate 'x', we need to divide both sides of the inequality by -7. Here, the critical rule regarding flipping the inequality sign comes into play. Since we are dividing by a negative number, we must reverse the inequality sign to maintain the integrity of the statement.

-7x > -17

Dividing both sides by -7, we get:

(-7x) / -7 < (-17) / -7

Observe that the '>' sign has been flipped to '<'. This simplifies to:

x < 17/7

3. Expressing the Solution Set

We have now arrived at the solution: x < 17/7. This inequality signifies that 'x' can assume any value less than 17/7. It's essential to recognize that this represents an infinite range of values. To visualize this solution, we can represent it on a number line. Draw a number line and mark the point 17/7. Since the inequality is 'less than' (and not 'less than or equal to'), we use an open circle at 17/7 to indicate that this value is not included in the solution. Then, we shade the region to the left of 17/7, representing all the values that satisfy the inequality.

Alternatively, we can express the solution in interval notation. Interval notation employs parentheses and brackets to represent ranges of values. Parentheses indicate that the endpoint is not included, while brackets indicate that it is included. For our solution, x < 17/7, the interval notation is (-∞, 17/7). The parenthesis next to -∞ indicates that negative infinity is not a specific number and is never included in an interval. The parenthesis next to 17/7 indicates that 17/7 is not included in the solution set.

4. Verifying the Solution's Validity

To ascertain the correctness of our solution, it's always prudent to verify it. We can achieve this by selecting a value within our solution range (x < 17/7) and substituting it back into the original inequality. If the inequality holds true, our solution is likely accurate. Let's choose x = 2 as a test value, since 2 is less than 17/7 (which is approximately 2.43).

Substituting x = 2 into the original inequality -7x + 9 > -8, we get:

-7(2) + 9 > -8

-14 + 9 > -8

-5 > -8

Navigating Potential Pitfalls and Common Errors

Solving inequalities, while conceptually akin to solving equations, presents certain unique challenges. One prevalent mistake is overlooking the need to flip the inequality sign when multiplying or dividing by a negative number. This can lead to an erroneous solution. Another pitfall lies in misinterpreting the inequality symbols. It's crucial to remember the distinction between '>', '<', '≥', and '≤' and to employ them correctly in expressing the solution.

Furthermore, when dealing with more intricate inequalities involving multiple steps, arithmetic errors can easily creep in. Therefore, it's imperative to perform each step meticulously and double-check your calculations. Consistent practice and a thorough comprehension of the underlying principles are paramount to avoiding these pitfalls and mastering the art of solving inequalities.

Conclusion: Empowering Your Inequality-Solving Prowess

In this comprehensive guide, we have meticulously dissected the process of solving the linear inequality -7x + 9 > -8. We commenced by establishing the fundamental principles of inequalities, emphasizing the cardinal rule of flipping the inequality sign when multiplying or dividing by a negative number. We then meticulously navigated each step of the solution, isolating the variable and expressing the solution in both inequality and interval notation.

Moreover, we underscored the significance of verifying the solution to ensure its accuracy and elucidated common pitfalls to circumvent. By internalizing these concepts and engaging in regular practice, you can cultivate a robust foundation in solving inequalities and confidently address a wide spectrum of mathematical problems. The ability to solve inequalities is not merely a valuable skill in mathematics but also a crucial asset in diverse fields, including science, engineering, and economics, where understanding relationships between quantities is of paramount importance.

Keywords: Solving inequalities, linear inequalities, step-by-step solution, flipping inequality sign, solution set, interval notation, verifying solutions, common mistakes, practice problems.