Solving Inequalities A Step-by-Step Guide For -26 + 13x + 2 > 2 - 13x

In this comprehensive guide, we will delve into the process of solving the inequality -26 + 13x + 2 > 2 - 13x for x. Inequalities, like equations, are mathematical statements that compare two expressions. However, instead of stating that the expressions are equal, inequalities use symbols like '>' (greater than), '<' (less than), '≥' (greater than or equal to), and '≤' (less than or equal to). Mastering the art of solving inequalities is crucial for various fields, including algebra, calculus, and real-world problem-solving. This article will provide a detailed, step-by-step solution, ensuring clarity and understanding for readers of all backgrounds. We will explore the fundamental principles behind manipulating inequalities, emphasizing the importance of maintaining the truth of the statement throughout the solving process. Understanding the nuances of inequality manipulation, such as how multiplication or division by a negative number affects the inequality sign, is key to arriving at the correct solution.

Before we dive into the solution, let's establish a solid foundation by understanding the basics of inequalities. An inequality compares two expressions using symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). The goal when solving an inequality is to isolate the variable on one side of the inequality sign, just like solving equations, but with some important differences. One key difference is that multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. For example, if we have -x > 2, multiplying both sides by -1 gives us x < -2. This is a critical rule to remember. Understanding this rule is crucial for solving inequalities correctly. In addition to understanding the basic rules, it's also important to be able to interpret the solution set. For example, if the solution to an inequality is x > 3, this means that any value of x greater than 3 will satisfy the original inequality. This solution set can be represented graphically on a number line, which provides a visual representation of all possible solutions. These fundamental concepts form the bedrock of solving inequalities, and a firm grasp of these principles will enable you to tackle more complex problems with confidence.

Let's solve the inequality -26 + 13x + 2 > 2 - 13x step by step. This will be a detailed walkthrough. Our initial inequality is -26 + 13x + 2 > 2 - 13x. The first step in solving this inequality, just like solving equations, is to simplify both sides of the inequality. This involves combining like terms. On the left side, we have -26 and +2, which are both constants. Adding these together gives us -24. So, the left side of the inequality simplifies to -24 + 13x. The right side of the inequality, 2 - 13x, is already in its simplest form, as there are no like terms to combine. So, after simplifying the left side, our inequality becomes -24 + 13x > 2 - 13x. This simplified form makes it easier to proceed with the next steps in isolating the variable x. Remember, the goal is to get x by itself on one side of the inequality, and we're making progress towards that with each step. Keep in mind the rules for manipulating inequalities, especially the rule about reversing the inequality sign when multiplying or dividing by a negative number. This will be crucial as we continue to solve for x.

Step 1: Simplify Both Sides

Combine the constant terms on the left side: -26 + 2 = -24. The inequality becomes -24 + 13x > 2 - 13x. Combining like terms is a fundamental step in simplifying any algebraic expression, whether it's part of an equation or an inequality. In this case, by combining the constants -26 and +2, we've made the inequality more manageable and easier to work with. This simplification allows us to focus on isolating the variable x in the subsequent steps. The process of combining like terms involves identifying terms that have the same variable raised to the same power (or are constants) and then adding or subtracting their coefficients. This step helps to reduce the complexity of the expression and makes it clearer to see the relationships between the different terms. It's important to pay close attention to the signs of the terms when combining them to ensure accuracy. A common mistake is to incorrectly combine terms with different signs, which can lead to an incorrect solution. So, always double-check your work when simplifying expressions.

Step 2: Group x Terms on One Side

Add 13x to both sides to get all x terms on the left: -24 + 13x + 13x > 2 - 13x + 13x. This simplifies to -24 + 26x > 2. The goal of this step is to consolidate all the terms containing the variable x onto one side of the inequality. By adding 13x to both sides, we eliminate the -13x term on the right side and move it to the left side, where it combines with the existing 13x term. This process maintains the balance of the inequality, as we are performing the same operation on both sides. Remember, whatever you do to one side of an inequality, you must do to the other side to preserve the relationship between the two expressions. After adding 13x to both sides and simplifying, the inequality is now in a more manageable form: -24 + 26x > 2. This sets the stage for the next step, which involves isolating the x term further by dealing with the constant term on the left side.

Step 3: Isolate the x Term

Add 24 to both sides to isolate the x term: -24 + 26x + 24 > 2 + 24. This simplifies to 26x > 26. Adding the same value to both sides of an inequality is a fundamental operation that preserves the inequality relationship. In this case, adding 24 to both sides cancels out the -24 on the left side, effectively isolating the term containing x. This step is crucial because it brings us closer to our ultimate goal of solving for x. By isolating the x term, we're essentially peeling away the layers of the inequality, one step at a time, until we have x by itself on one side. After this step, the inequality becomes 26x > 26, which is a much simpler form to work with. It's important to perform this step carefully, ensuring that you add the correct value to both sides and simplify the resulting expression accurately. This meticulous approach is key to avoiding errors and arriving at the correct solution.

Step 4: Solve for x

Divide both sides by 26 to solve for x: (26x) / 26 > 26 / 26. This simplifies to x > 1. This is the final step in solving the inequality, and it involves isolating x completely. To do this, we divide both sides of the inequality by the coefficient of x, which is 26 in this case. Dividing both sides by the same positive number preserves the direction of the inequality. It's important to remember that if we were dividing by a negative number, we would need to reverse the inequality sign. However, since we're dividing by a positive number, we don't need to worry about that here. After dividing both sides by 26, we arrive at the solution: x > 1. This means that any value of x that is greater than 1 will satisfy the original inequality. This solution can be visualized on a number line as an open interval extending to the right from 1. Understanding how to interpret the solution set is just as important as the steps involved in solving the inequality itself. So, make sure you understand what the solution x > 1 means in the context of the original problem.

Therefore, the solution to the inequality -26 + 13x + 2 > 2 - 13x is x > 1. This means any value of x greater than 1 will satisfy the original inequality. To verify this solution, we can choose a value greater than 1, such as x = 2, and substitute it into the original inequality. This will give us -26 + 13(2) + 2 > 2 - 13(2), which simplifies to -26 + 26 + 2 > 2 - 26, and further simplifies to 2 > -24. This is a true statement, which confirms that our solution x > 1 is correct. Similarly, we can choose a value less than or equal to 1, such as x = 1, and substitute it into the original inequality. This will give us -26 + 13(1) + 2 > 2 - 13(1), which simplifies to -26 + 13 + 2 > 2 - 13, and further simplifies to -11 > -11. This is a false statement, which further confirms that our solution x > 1 is correct. Understanding how to verify your solution is a crucial skill in mathematics, as it allows you to check your work and ensure that you have arrived at the correct answer. It also helps to solidify your understanding of the concepts involved.

To verify the solution, we can substitute a value of x greater than 1 into the original inequality. Let's use x = 2: -26 + 13(2) + 2 > 2 - 13(2). This simplifies to -26 + 26 + 2 > 2 - 26, which further simplifies to 2 > -24. This is true, confirming our solution. Verifying the solution is a critical step in the problem-solving process, as it helps to ensure that the answer obtained is correct and satisfies the original conditions of the problem. In the context of inequalities, verification involves substituting a value within the solution set back into the original inequality to see if it holds true. If the inequality holds true, it provides evidence that the solution set is correct. However, it's important to note that verifying one value does not guarantee the correctness of the entire solution set. To be completely sure, it's often helpful to test multiple values within the solution set, as well as values outside the solution set, to confirm that they do not satisfy the inequality. This process provides a more comprehensive check of the solution and can help to identify any potential errors. In this case, by substituting x = 2 into the original inequality, we were able to confirm that our solution x > 1 is indeed correct.

In this article, we have provided a detailed, step-by-step solution to the inequality -26 + 13x + 2 > 2 - 13x. We have covered the simplification process, the importance of maintaining the inequality's balance, and the crucial step of verifying the solution. By following these steps, you can confidently solve similar inequalities. Solving inequalities is a fundamental skill in mathematics, and mastering this skill opens the door to tackling more complex problems in algebra, calculus, and other areas of mathematics. The ability to manipulate inequalities, understand the rules that govern them, and interpret the solution sets is essential for success in these fields. Furthermore, the problem-solving strategies and logical thinking skills that you develop by working with inequalities are valuable assets that can be applied to a wide range of real-world situations. So, continue to practice solving inequalities and building your mathematical skills, and you'll be well-equipped to tackle any challenge that comes your way. Remember, the key to success in mathematics is consistent practice and a willingness to learn from your mistakes.