Solving Inequalities A Step-by-Step Guide For 22q + 73 > 52q + 63

In the realm of mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Solving inequalities, much like solving equations, involves isolating the variable of interest to determine the range of values that satisfy the given condition. This comprehensive guide will delve into the process of solving the inequality 22q + 73 > 52q + 63, providing a step-by-step approach to arrive at the solution. Understanding inequalities is fundamental for various mathematical concepts and real-world applications. They help us define boundaries, constraints, and ranges, making them indispensable in fields like optimization, economics, and computer science. In this article, we will not only solve the specific inequality but also discuss the underlying principles and techniques that can be applied to a wide range of inequality problems. By mastering these concepts, you will be equipped to tackle more complex mathematical challenges and gain a deeper appreciation for the power and versatility of inequalities. Let's embark on this mathematical journey and unravel the intricacies of solving for q in the given inequality. We will break down each step, ensuring clarity and comprehension, so that you can confidently apply these methods to future problems. Inequalities, while similar to equations, have unique properties that must be considered. For instance, multiplying or dividing an inequality by a negative number requires flipping the inequality sign. This is a crucial rule to remember to avoid errors in your solutions. Throughout this guide, we will highlight such important nuances and provide examples to illustrate their application. So, whether you are a student learning the basics or someone looking to refresh your knowledge, this guide will provide you with the tools and understanding necessary to solve inequalities effectively.

Step 1: Isolate the Variable Terms

The primary goal in solving any inequality is to isolate the variable on one side of the inequality sign. In this case, we want to isolate q. To begin, we need to gather all the terms containing q on one side of the inequality. A strategic approach is to move the term with the smaller coefficient of q to the side with the larger coefficient. This minimizes the chances of dealing with negative coefficients, which can introduce additional steps and potential for errors. In our inequality, 22q + 73 > 52q + 63, we observe that 22q has a smaller coefficient than 52q. Therefore, we will subtract 22q from both sides of the inequality. This maintains the balance of the inequality, ensuring that the relationship between the two sides remains valid. Subtracting 22q from both sides, we get:

22q + 73 - 22q > 52q + 63 - 22q

Simplifying this expression, the 22q terms on the left side cancel out, leaving us with:

73 > 30q + 63

Now, we have successfully moved all the terms containing q to the right side of the inequality. The next step involves isolating the constant terms on the other side, which we will address in the subsequent section. Remember, the key to isolating the variable is to perform the same operations on both sides of the inequality, ensuring that the balance is maintained. This principle applies to addition, subtraction, multiplication, and division, with the crucial exception of multiplying or dividing by a negative number, which requires flipping the inequality sign. This step-by-step approach allows us to systematically simplify the inequality and move closer to the solution. By understanding the rationale behind each step, you can confidently apply these techniques to a variety of inequality problems. The goal is not just to arrive at the correct answer but to also understand the underlying process and the principles that govern it. This deeper understanding will empower you to tackle more complex mathematical challenges and develop a strong foundation in algebra.

Step 2: Isolate the Constant Terms

Having successfully moved the variable terms to one side, our next objective is to isolate the constant terms on the opposite side of the inequality. In our current state, the inequality is 73 > 30q + 63. We need to eliminate the constant term 63 from the right side of the inequality. To achieve this, we will subtract 63 from both sides of the inequality. This operation maintains the balance of the inequality, ensuring that the relationship between the two sides remains consistent. Subtracting 63 from both sides, we get:

73 - 63 > 30q + 63 - 63

Simplifying this expression, the 63 terms on the right side cancel out, leaving us with:

10 > 30q

Now, we have successfully isolated the constant terms on the left side of the inequality. The next step involves isolating the variable q by eliminating its coefficient. This will reveal the range of values that q can take to satisfy the inequality. Remember, the key to isolating the variable is to perform inverse operations. We subtracted 63 because it was added to the term containing q. Similarly, we will use division to eliminate the coefficient of q. This systematic approach ensures that we are moving closer to the solution with each step. Inequalities, like equations, require careful attention to detail. Each operation must be performed accurately and consistently to avoid errors. The goal is not just to find the solution but to also understand the process and the reasoning behind each step. This deeper understanding will empower you to tackle more complex mathematical problems and develop a strong foundation in algebra. As we proceed to the next step, we will continue to emphasize the importance of maintaining balance and performing inverse operations to isolate the variable of interest.

Step 3: Solve for q

With the inequality simplified to 10 > 30q, our final step is to isolate q completely. Currently, q is being multiplied by 30. To undo this multiplication, we need to perform the inverse operation, which is division. We will divide both sides of the inequality by 30. It's crucial to remember that when dividing (or multiplying) an inequality by a negative number, we must flip the inequality sign. However, in this case, we are dividing by a positive number (30), so we do not need to flip the sign. Dividing both sides by 30, we get:

10 / 30 > 30q / 30

Simplifying this expression, we have:

1/3 > q

This inequality can also be written as:

q < 1/3

This is our solution. It states that q can be any value less than 1/3. To ensure our solution is in the simplest form, we have reduced the fraction 10/30 to its lowest terms, which is 1/3. The solution q < 1/3 represents a range of values, not just a single value. This is a fundamental difference between solving inequalities and solving equations. Inequalities often have an infinite number of solutions, while equations typically have a finite number of solutions. To fully understand the solution, it's helpful to visualize it on a number line. The solution q < 1/3 would be represented by an open circle at 1/3 and a line extending to the left, indicating all values less than 1/3. This visual representation provides a clear picture of the solution set. In conclusion, by systematically isolating the variable q, we have successfully solved the inequality 22q + 73 > 52q + 63. The solution, q < 1/3, represents all values of q that satisfy the given condition. This process highlights the importance of performing inverse operations and maintaining balance when working with inequalities. With practice and a solid understanding of these principles, you can confidently solve a wide range of inequality problems.

Final Answer

The solution to the inequality 22q + 73 > 52q + 63 is:

q < 1/3

This means that any value of q that is less than 1/3 will satisfy the original inequality. We have arrived at this solution by following a step-by-step process, which included isolating the variable terms, isolating the constant terms, and finally, solving for q. Each step was performed carefully, ensuring that the balance of the inequality was maintained and that the operations were performed correctly. The fraction 1/3 is already in its lowest terms, as required by the problem statement. We did not round our answer or use mixed fractions, adhering to the given instructions. To verify our solution, we can substitute a value less than 1/3 into the original inequality and check if it holds true. For example, let's choose q = 0. Substituting this value into the original inequality, we get:

22(0) + 73 > 52(0) + 63

73 > 63

This is true, which confirms that our solution is correct. Similarly, we can substitute a value greater than 1/3 and check if the inequality does not hold true. This provides further validation of our solution. In summary, we have successfully solved the inequality and expressed the solution in the required format. The process involved a clear understanding of inequality properties and the application of inverse operations. With this understanding, you can confidently tackle similar problems and apply these techniques to more complex mathematical challenges.