Solving Inequalities Finding Solutions For 10(3x+2) > 7(2x-4)
Inequalities, in the realm of mathematics, serve as powerful tools for expressing relationships where values are not necessarily equal. Unlike equations that pinpoint exact solutions, inequalities define a range of possible values that satisfy a given condition. This exploration delves into the fascinating world of inequalities, equipping you with the knowledge and skills to confidently tackle them. At the heart of this discussion is a specific inequality: 10(3x + 2) > 7(2x - 4). We will meticulously dissect this inequality, revealing the step-by-step process of finding its solution set and ultimately determining the correct answer from a set of options. Mastering the art of solving inequalities is not merely an academic exercise; it's a fundamental skill that permeates various fields, from economics and finance to engineering and computer science. Understanding how to manipulate inequalities allows us to model real-world scenarios, make informed decisions, and optimize outcomes. So, embark on this journey with us as we unravel the intricacies of inequalities and empower you to conquer them with ease. Whether you are a student grappling with algebraic concepts or a professional seeking to refine your analytical prowess, this guide will serve as your trusted companion in the realm of inequalities.
Understanding Inequalities
Before we plunge into the specifics of the problem at hand, let's establish a firm grasp of the fundamental concepts surrounding inequalities. Inequalities are mathematical statements that compare two expressions using inequality symbols. These symbols include:
- > (greater than)
- < (less than)
- ≥ (greater than or equal to)
- ≤ (less than or equal to)
The solutions to an inequality are all the values that make the inequality true. Unlike equations, which typically have a finite number of solutions, inequalities often have an infinite number of solutions, representing a range of values. This range can be visualized on a number line, where the solutions are represented by an interval. The interval may be open (excluding the endpoints) or closed (including the endpoints), depending on the inequality symbol used. For instance, x > 2 represents all values greater than 2, excluding 2 itself, while x ≥ 2 includes 2 in the solution set. The ability to interpret and manipulate inequalities is crucial for problem-solving in various contexts. Understanding the properties of inequalities, such as the rules for adding, subtracting, multiplying, and dividing both sides, is essential for isolating the variable and determining the solution set. Furthermore, recognizing the impact of these operations on the inequality symbol, especially when multiplying or dividing by a negative number, is paramount to avoid errors. With a solid understanding of these fundamental concepts, we are well-equipped to tackle the complexities of more challenging inequalities.
Step-by-Step Solution
To solve the inequality 10(3x + 2) > 7(2x - 4), we'll follow a series of algebraic steps to isolate the variable x. Let's break down the process:
1. Distribute
The first step involves distributing the constants on both sides of the inequality. This means multiplying the numbers outside the parentheses by each term inside the parentheses:
10 * (3x + 2) > 7 * (2x - 4)
30x + 20 > 14x - 28
This distribution eliminates the parentheses, making the inequality easier to manipulate. By applying the distributive property, we transform the original inequality into a more manageable form, setting the stage for subsequent steps. This step is crucial for simplifying the inequality and bringing it closer to a form where the variable can be isolated. The accuracy of this step is paramount, as any errors in distribution will propagate through the rest of the solution. Therefore, careful attention to detail is essential to ensure the correct application of the distributive property. With the parentheses removed, we can now proceed to combine like terms and isolate the variable, ultimately leading us to the solution set of the inequality.
2. Combine Like Terms
Next, we want to gather the x terms on one side of the inequality and the constant terms on the other side. To do this, we'll subtract 14x from both sides:
30x + 20 - 14x > 14x - 28 - 14x
16x + 20 > -28
Now, subtract 20 from both sides:
16x + 20 - 20 > -28 - 20
16x > -48
This process of combining like terms streamlines the inequality, bringing us closer to isolating the variable x. By strategically adding or subtracting terms from both sides, we maintain the balance of the inequality while consolidating similar terms. This step is crucial for simplifying the expression and making it easier to solve for the unknown variable. The goal is to isolate the term containing x on one side of the inequality, while gathering the constant terms on the other side. This rearrangement facilitates the final step of dividing by the coefficient of x, which will reveal the solution set. Careful attention to the signs of the terms is essential during this process, as errors in addition or subtraction can lead to an incorrect solution.
3. Isolate the Variable
Finally, to isolate x, we'll divide both sides of the inequality by 16:
16x / 16 > -48 / 16
x > -3
This step completes the process of solving the inequality. By dividing both sides by the coefficient of x, we unveil the solution set, which represents all values of x that satisfy the original inequality. It's crucial to remember that when dividing both sides of an inequality by a negative number, the inequality sign must be flipped. However, in this case, we are dividing by a positive number (16), so the inequality sign remains unchanged. The solution x > -3 indicates that any value of x greater than -3 will make the original inequality true. This solution can be visualized on a number line, where an open circle at -3 represents the exclusion of -3 from the solution set, and an arrow extending to the right indicates all values greater than -3. This final step is the culmination of the previous steps, and it provides the answer to the problem. Verifying the solution by substituting values greater than -3 into the original inequality can further solidify understanding and confirm the accuracy of the result.
Identifying the Correct Answer
Now that we have solved the inequality, we can identify the correct answer from the given options.
The solution we found is x > -3. Comparing this to the options provided:
- A. x > -4
- B. x < -4
- C. x > -3
- D. x < -3
It's clear that option C. x > -3 matches our solution.
Therefore, the correct answer is C. This step involves carefully comparing the solution obtained through algebraic manipulation with the provided options. It requires a clear understanding of the inequality symbols and their meanings. In this case, the solution x > -3 represents all values of x that are strictly greater than -3. Option C accurately reflects this solution, while the other options represent different ranges of values. Option A, x > -4, includes values that are less than -3, which do not satisfy the original inequality. Option B, x < -4, represents values that are significantly smaller than -3 and would not make the inequality true. Option D, x < -3, represents values that are less than -3, which also do not satisfy the original inequality. Therefore, by carefully comparing the solution with each option, we can confidently identify the correct answer as C, x > -3. This process reinforces the importance of accurate algebraic manipulation and a thorough understanding of inequality symbols.
Common Mistakes to Avoid
When solving inequalities, it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:
1. 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. For example, if you have -2x > 4, dividing both sides by -2 requires flipping the sign: x < -2.
2. Incorrectly Distributing
Another common error is distributing incorrectly. Make sure to multiply the constant outside the parentheses by every term inside the parentheses. For instance, in the expression 3(x + 2), you should distribute the 3 to both the x and the 2, resulting in 3x + 6, not just 3x + 2.
3. Arithmetic Errors
Simple arithmetic errors can also lead to incorrect solutions. Double-check your calculations, especially when dealing with negative numbers. A small mistake in addition, subtraction, multiplication, or division can significantly impact the final result.
4. Misinterpreting the Inequality Symbols
Understanding the meaning of the inequality symbols is crucial. > means greater than, < means less than, ≥ means greater than or equal to, and ≤ means less than or equal to. Misinterpreting these symbols can lead to selecting the wrong solution set.
5. Not Checking the Solution
Always check your solution by plugging a value from your solution set back into the original inequality. This helps ensure that your answer is correct and can catch any errors you may have made during the solving process.
By being aware of these common mistakes and taking the time to double-check your work, you can significantly improve your accuracy when solving inequalities.
Conclusion
In conclusion, solving inequalities requires a systematic approach and a careful understanding of algebraic principles. By following the steps of distributing, combining like terms, and isolating the variable, we can effectively determine the solution set. Remember to pay close attention to the inequality sign, especially when multiplying or dividing by a negative number. Avoiding common mistakes and double-checking your work are essential for accuracy. Mastering inequalities is a valuable skill that extends beyond the classroom, with applications in various fields. The specific inequality we addressed, 10(3x + 2) > 7(2x - 4), serves as a practical example of the problem-solving process. By breaking down the problem into manageable steps, we were able to confidently arrive at the correct solution: x > -3. This solution represents all values of x that satisfy the given inequality. Furthermore, understanding the common mistakes associated with solving inequalities, such as forgetting to flip the inequality sign or incorrectly distributing, can help prevent errors and ensure accurate results. By consistently applying these strategies, you can enhance your ability to solve inequalities and tackle more complex mathematical challenges. The journey of mastering inequalities is an ongoing process, and with practice and perseverance, you can unlock the power of this fundamental mathematical tool.
By understanding the underlying concepts and practicing regularly, you can confidently solve inequalities and apply them to real-world problems.
