Solving (-5x + 4) / (3x + 3) > -11/12 Inequality With Interval Notation
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In this comprehensive guide, we will delve into the step-by-step process of solving the inequality (-5x + 4) / (3x + 3) > -11/12. This type of problem is common in algebra and calculus, and mastering it requires a solid understanding of algebraic manipulations and interval notation. We will break down each step in detail, ensuring clarity and providing explanations along the way. By the end of this article, you will not only be able to solve this specific inequality but also gain the confidence to tackle similar problems. Let’s dive in and explore the world of inequalities!
1. Introduction to Inequalities
Before we tackle the main problem, let's briefly discuss what inequalities are and why they are important in mathematics. Inequalities are mathematical expressions that show the relationship between two values that are not necessarily equal. They are used extensively in various fields, including economics, physics, and computer science, to model and solve real-world problems. Understanding how to solve inequalities is crucial for anyone studying mathematics or related disciplines. The solution to an inequality is often represented as an interval or a union of intervals, which we will cover in detail later in this guide. Inequalities, unlike equations, can have a range of solutions, making them versatile tools for modeling real-world scenarios where exact values are not always attainable or necessary. From determining the range of profitable prices for a product to calculating the minimum speed required to escape Earth's gravity, inequalities provide a framework for understanding and solving complex problems. In essence, mastering inequalities opens up a world of mathematical possibilities and enhances problem-solving skills across various domains.
2. Understanding the Problem: (-5x + 4) / (3x + 3) > -11/12
Now, let's take a closer look at the inequality we aim to solve: (-5x + 4) / (3x + 3) > -11/12. This is a rational inequality, meaning it involves a rational expression (a fraction with polynomials in the numerator and denominator). Solving rational inequalities requires a slightly different approach compared to solving linear or quadratic inequalities. The key is to eliminate the fraction by multiplying both sides by a common denominator, but we must be cautious about the sign of the denominator, as it can affect the direction of the inequality. In this particular inequality, we have a linear expression in both the numerator and the denominator. The presence of the variable x in both parts of the fraction means that the solution will likely be a range of values, rather than a single point. Furthermore, the inequality sign (>) indicates that we are looking for values of x that make the left-hand side strictly greater than -11/12. Understanding these nuances is crucial for choosing the correct solution strategy and interpreting the final result. The goal is to isolate x and determine the interval(s) where the inequality holds true.
3. Step-by-Step Solution
To solve the inequality (-5x + 4) / (3x + 3) > -11/12, we will follow these steps:
3.1. Eliminate the Fraction
The first step in solving this rational inequality is to eliminate the fractions. To do this, we need to multiply both sides of the inequality by the least common multiple (LCM) of the denominators. In this case, the denominators are (3x + 3) and 12. The LCM of these is 12(3x + 3). However, we need to be careful because the sign of (3x + 3) can change the direction of the inequality. So, let's first factor out the 3 from (3x + 3) to get 3(x + 1). Now our denominators are 3(x + 1) and 12, and their LCM is 12(x + 1). We will consider two cases:
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Case 1: x + 1 > 0 (i.e., x > -1)
If x + 1 > 0, then 12(x + 1) is positive, and we can multiply both sides of the inequality by 12(x + 1) without changing the direction of the inequality sign:
12(x + 1) * [(-5x + 4) / (3(x + 1))] > 12(x + 1) * (-11/12)
Simplifying, we get:
4(-5x + 4) > -11(x + 1)
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Case 2: x + 1 < 0 (i.e., x < -1)
If x + 1 < 0, then 12(x + 1) is negative, and we must reverse the direction of the inequality sign when multiplying:
12(x + 1) * [(-5x + 4) / (3(x + 1))] < 12(x + 1) * (-11/12)
Simplifying, we get:
4(-5x + 4) < -11(x + 1)
3.2. Simplify and Solve for x (Case 1: x > -1)
Let's continue with Case 1, where x > -1. We have the inequality:
4(-5x + 4) > -11(x + 1)
Expanding both sides, we get:
-20x + 16 > -11x - 11
Now, we want to isolate x. Add 20x to both sides:
16 > 9x - 11
Add 11 to both sides:
27 > 9x
Divide both sides by 9:
3 > x
So, in this case, we have x < 3. Combining this with our initial condition of x > -1, we get the interval -1 < x < 3.
3.3. Simplify and Solve for x (Case 2: x < -1)
Now, let's consider Case 2, where x < -1. We have the inequality:
4(-5x + 4) < -11(x + 1)
Expanding both sides, we get:
-20x + 16 < -11x - 11
Add 20x to both sides:
16 < 9x - 11
Add 11 to both sides:
27 < 9x
Divide both sides by 9:
3 < x
So, in this case, we have x > 3. However, this contradicts our initial condition of x < -1. Therefore, there is no solution in this case.
3.4. Combine the Solutions
From Case 1, we found that -1 < x < 3. Case 2 yielded no solutions. Therefore, the solution to the inequality is -1 < x < 3.
4. Expressing the Solution in Interval Notation
Now that we have found the solution -1 < x < 3, we need to express it using interval notation. Interval notation is a way of writing sets of real numbers using intervals. It uses parentheses and brackets to indicate whether the endpoints are included in the set. A parenthesis indicates that the endpoint is not included (open interval), while a bracket indicates that the endpoint is included (closed interval). Since our solution is -1 < x < 3, which means x is strictly greater than -1 and strictly less than 3, we will use parentheses. Therefore, the solution in interval notation is (-1, 3). This notation succinctly represents all the values of x that satisfy the original inequality. In this interval, any number between -1 and 3 (excluding -1 and 3 themselves) will make the inequality (-5x + 4) / (3x + 3) > -11/12 true. Interval notation is a standard way to express solutions to inequalities in mathematics, providing a clear and concise representation of the range of possible values.
5. Graphical Representation of the Solution
Visualizing the solution on a number line can provide a clearer understanding of the interval notation. To represent the solution (-1, 3) graphically, draw a number line and mark the points -1 and 3. Since the interval is open (using parentheses), we will use open circles at -1 and 3 to indicate that these points are not included in the solution. Then, shade the region between -1 and 3 to represent all the values of x that satisfy the inequality. This shaded region visually represents the infinite number of values between -1 and 3 that make the inequality true. The open circles at the endpoints serve as a reminder that the solution does not include -1 and 3. This graphical representation complements the interval notation and provides a visual confirmation of the solution set. It's a helpful tool for understanding inequalities and their solutions, especially when dealing with more complex inequalities or systems of inequalities.
6. Checking the Solution
To ensure the accuracy of our solution, it's always a good practice to check it. We can do this by selecting a test value within the interval (-1, 3) and substituting it into the original inequality. If the inequality holds true for the test value, it strengthens our confidence in the solution. Let's choose x = 0 as our test value, since 0 lies within the interval (-1, 3). Substituting x = 0 into the original inequality, we get:
(-5(0) + 4) / (3(0) + 3) > -11/12
Simplifying, we have:
4 / 3 > -11/12
To compare these fractions, we can find a common denominator, which is 12. So, we rewrite 4/3 as 16/12:
16/12 > -11/12
This inequality is clearly true, as 16/12 is greater than -11/12. This confirms that our solution interval (-1, 3) is likely correct. While a single test value doesn't guarantee the solution is perfect, it provides a good indication of its validity. For further assurance, one could test additional values within the interval or even values outside the interval to confirm they do not satisfy the inequality. This process of verification is a crucial step in problem-solving, especially when dealing with inequalities or equations.
7. Common Mistakes to Avoid
When solving inequalities, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct solution. One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This is a critical step in solving inequalities, and overlooking it can lead to an incorrect solution. Another common mistake is failing to consider the domain of the variable, especially in rational inequalities. In our problem, we had to consider the cases where (3x + 3) was positive and negative, as this affected the direction of the inequality. Additionally, students may sometimes make errors in algebraic manipulations, such as incorrectly expanding expressions or combining like terms. Careful attention to detail and double-checking each step can help prevent these errors. Another pitfall is not expressing the solution in the correct notation, such as using brackets instead of parentheses or vice versa. Understanding the difference between open and closed intervals is crucial for accurately representing the solution set. By being mindful of these common mistakes and practicing careful problem-solving techniques, you can significantly improve your accuracy and confidence in solving inequalities.
8. Conclusion
In this comprehensive guide, we have thoroughly explored the process of solving the inequality (-5x + 4) / (3x + 3) > -11/12. We began by understanding the problem and outlining the necessary steps. We then carefully eliminated the fractions, considered the different cases based on the sign of the denominator, and solved for x in each case. After obtaining the solution, we expressed it using interval notation and provided a graphical representation for better understanding. We also emphasized the importance of checking the solution to ensure accuracy and highlighted common mistakes to avoid. By following these steps and practicing similar problems, you can develop a strong understanding of how to solve rational inequalities. Remember, the key to success in mathematics is consistent practice and a willingness to learn from mistakes. Solving inequalities is a fundamental skill in algebra and calculus, and mastering it will open doors to more advanced mathematical concepts and applications. We hope this guide has been helpful in your journey to understanding and solving inequalities.
