Solving 2(4 + 2x) ≥ 5x + 5 A Comprehensive Guide
#title: Solving Inequalities A Step-by-Step Guide to 2(4 + 2x) ≥ 5x + 5
In this comprehensive guide, we will walk through the process of solving the inequality 2(4 + 2x) ≥ 5x + 5. Inequalities, a fundamental concept in algebra, express the relative order of two values or expressions. Unlike equations, which state that two expressions are equal, inequalities use symbols like '≥' (greater than or equal to), '≤' (less than or equal to), '>' (greater than), and '<' (less than) to show the relationship between quantities. Mastering the techniques for solving inequalities is crucial for various applications in mathematics, science, and engineering. From determining the range of possible solutions to optimizing real-world scenarios, the ability to manipulate and interpret inequalities is an invaluable skill. This article will break down the steps involved in solving the given inequality, providing clear explanations and examples to ensure a thorough understanding. We will begin by simplifying the inequality through distribution and combining like terms, then isolate the variable 'x' to find the solution set. This step-by-step approach will not only solve the specific problem but also equip you with the knowledge to tackle a wide range of inequality problems. Whether you are a student learning algebra or someone looking to refresh your mathematical skills, this guide offers a clear and concise pathway to understanding and solving inequalities. By the end of this article, you will confidently apply these techniques to solve similar problems and appreciate the role of inequalities in mathematical problem-solving.
Understanding the Problem
Before diving into the solution, let's fully understand the inequality we're dealing with: 2(4 + 2x) ≥ 5x + 5. This expression states that the value of 2 multiplied by the quantity (4 + 2x) is greater than or equal to the sum of 5x and 5. Our goal is to find all values of 'x' that satisfy this condition. Inequalities are used extensively in mathematics and real-world applications to represent constraints, ranges, and comparisons. They are essential in fields such as optimization, where we seek to maximize or minimize quantities subject to certain limitations, and in statistics, where they are used to define confidence intervals and hypothesis tests. In this particular inequality, we have a linear inequality, which means that the highest power of the variable 'x' is 1. Linear inequalities can be solved using algebraic manipulations similar to those used for linear equations, but with an important distinction: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. This is because multiplying or dividing by a negative number changes the order of the numbers on the number line. Understanding the structure of the inequality is crucial for choosing the correct solution strategy. We must first simplify the expression by distributing the 2 on the left side and then combine like terms to isolate the variable 'x'. This process involves careful attention to detail and a solid understanding of the properties of inequalities. By breaking down the problem into smaller, manageable steps, we can arrive at the solution systematically and accurately. The ability to interpret and solve inequalities is a fundamental skill in algebra, and mastering it will provide a strong foundation for more advanced mathematical concepts. So, let's begin by simplifying the inequality and moving closer to finding the values of 'x' that satisfy the given condition.
Step 1: Distribute on the Left Side
To begin solving the inequality, our first step is to distribute the 2 on the left side of the inequality: 2(4 + 2x) ≥ 5x + 5. Distribution involves multiplying the term outside the parentheses by each term inside the parentheses. In this case, we multiply 2 by both 4 and 2x. This process is based on the distributive property of multiplication over addition, which states that a(b + c) = ab + ac. Applying this property to our inequality, we get: 2 * 4 + 2 * 2x ≥ 5x + 5. Performing the multiplications, we have: 8 + 4x ≥ 5x + 5. Now, the inequality is simplified, and we have eliminated the parentheses. This step is crucial because it allows us to combine like terms and isolate the variable 'x'. The distributive property is a fundamental concept in algebra and is used extensively in simplifying expressions and solving equations and inequalities. It ensures that the multiplication is carried out correctly across the addition or subtraction within the parentheses. By distributing the 2, we have transformed the inequality into a more manageable form, making it easier to proceed with the next steps in the solution. The simplified inequality, 8 + 4x ≥ 5x + 5, now has terms that can be combined, which will help us isolate 'x' and determine the range of values that satisfy the inequality. This step-by-step approach is essential for solving inequalities accurately and efficiently. Each step builds upon the previous one, leading us closer to the solution. Now that we have distributed and simplified the left side, we can move on to the next step, which involves combining like terms and isolating the variable 'x'. This will bring us closer to finding the solution set for the inequality.
Step 2: Combine Like Terms
After distributing, we have the inequality 8 + 4x ≥ 5x + 5. The next step in solving this inequality is to combine like terms. This involves grouping terms with the same variable (in this case, 'x') and constant terms together. Our goal is to isolate the variable 'x' on one side of the inequality. To do this, we can subtract 4x from both sides of the inequality. Subtracting the same value from both sides of an inequality maintains the inequality's balance, just as it does in an equation. This gives us: 8 + 4x - 4x ≥ 5x - 4x + 5. Simplifying, we get: 8 ≥ x + 5. Now, we need to isolate 'x' further by subtracting 5 from both sides of the inequality: 8 - 5 ≥ x + 5 - 5. This simplifies to: 3 ≥ x. Combining like terms is a critical step in solving inequalities and equations. It helps to simplify the expression and make it easier to isolate the variable. By subtracting 4x from both sides, we eliminated the 'x' term from the left side, bringing us closer to isolating 'x'. Similarly, subtracting 5 from both sides will isolate 'x' completely. This process relies on the properties of inequalities, which allow us to perform the same operations on both sides without changing the validity of the inequality, as long as we do not multiply or divide by a negative number. The inequality 3 ≥ x means that 3 is greater than or equal to x. This can also be written as x ≤ 3, which is often a more intuitive way to express the solution. This inequality represents a range of values for 'x' that satisfy the original inequality. Now that we have isolated 'x', we have found the solution set. The next step is to interpret and express this solution in a clear and concise manner.
Step 3: Isolate x
We've reached the point where we have the inequality 3 ≥ x. This inequality directly tells us the solution set, but it's important to isolate x and express the solution in a standard form. The inequality 3 ≥ x is equivalent to x ≤ 3. This means that 'x' can be any value that is less than or equal to 3. Isolating 'x' in this way provides a clear and concise understanding of the solution set. In mathematical notation, the solution set can be represented in several ways. One common way is to use interval notation. The interval notation for x ≤ 3 is (-∞, 3]. The parenthesis '(' indicates that negative infinity is not included in the solution set (as infinity is not a specific number), while the bracket ']' indicates that 3 is included in the solution set. Another way to represent the solution set is graphically. On a number line, we would draw a closed circle (or a filled-in dot) at 3, indicating that 3 is included in the solution, and then shade the line to the left of 3, representing all values less than 3. Understanding how to isolate the variable and express the solution set in different forms is crucial for interpreting and communicating mathematical results. The ability to translate between different representations, such as inequalities, interval notation, and graphical representations, is a valuable skill in mathematics. By isolating 'x', we have not only found the solution but also made it easier to understand and apply. The solution x ≤ 3 provides a complete picture of the values that satisfy the original inequality. Now, we can confidently state the solution and move on to verifying our answer to ensure its accuracy. This step-by-step process of isolating the variable is a fundamental technique in solving inequalities and equations, and mastering it will greatly enhance your problem-solving abilities in mathematics.
Step 4: Express the Solution
Now that we have isolated 'x' and found that x ≤ 3, the next important step is to express the solution in a clear and understandable manner. The inequality x ≤ 3 indicates that the solution set includes all real numbers that are less than or equal to 3. This means that any value of 'x' that is 3 or smaller will satisfy the original inequality 2(4 + 2x) ≥ 5x + 5. To express this solution set comprehensively, we can use different notations, each providing a unique way to visualize and understand the solution. One common method is using interval notation, as mentioned earlier. In interval notation, the solution x ≤ 3 is written as (-∞, 3]. This notation clearly shows that the solution includes all numbers from negative infinity up to and including 3. The parenthesis '(' next to negative infinity signifies that negative infinity is not a specific number and is therefore not included in the set, while the bracket ']' next to 3 indicates that 3 is included in the set. Another way to express the solution is graphically, using a number line. To represent x ≤ 3 on a number line, we would draw a number line and mark the point 3. Since the inequality includes 'equal to', we would use a closed circle (or a filled-in dot) at 3 to indicate that 3 is part of the solution. Then, we would shade the portion of the number line to the left of 3, representing all numbers less than 3. This graphical representation provides a visual understanding of the solution set. In addition to interval notation and graphical representation, the solution can also be expressed in words. We can simply state that the solution set consists of all real numbers less than or equal to 3. Expressing the solution in multiple ways helps to reinforce understanding and allows for effective communication of the result. Each notation provides a different perspective on the solution set, making it easier to grasp the range of values that satisfy the inequality. The ability to express solutions in various forms is a valuable skill in mathematics, as it allows for clear and concise communication of mathematical ideas. Now that we have expressed the solution in multiple ways, we can be confident that we have a thorough understanding of the solution set for the inequality.
Step 5: Verify the Solution
After finding the solution to an inequality, it is crucial to verify the solution to ensure its accuracy. This step helps to catch any potential errors made during the solving process and confirms that the solution set we obtained is indeed correct. To verify the solution x ≤ 3, we can test values within the solution set and values outside the solution set in the original inequality 2(4 + 2x) ≥ 5x + 5. If our solution is correct, values within the solution set should satisfy the inequality, while values outside the solution set should not. Let's first test a value within the solution set, such as x = 0. Substituting x = 0 into the original inequality, we get: 2(4 + 2(0)) ≥ 5(0) + 5. This simplifies to: 2(4) ≥ 5, which further simplifies to: 8 ≥ 5. This statement is true, so x = 0 satisfies the inequality, which supports our solution. Now, let's test a value outside the solution set, such as x = 4. Substituting x = 4 into the original inequality, we get: 2(4 + 2(4)) ≥ 5(4) + 5. This simplifies to: 2(4 + 8) ≥ 20 + 5, which further simplifies to: 2(12) ≥ 25, resulting in: 24 ≥ 25. This statement is false, so x = 4 does not satisfy the inequality, which also supports our solution. By testing values within and outside the solution set, we have gained confidence that our solution x ≤ 3 is correct. Verification is an essential step in problem-solving, as it helps to identify and correct errors. It also reinforces understanding of the solution set and the inequality itself. In addition to testing specific values, we can also verify the solution by reviewing the steps we took to solve the inequality, ensuring that each step was performed correctly and that no algebraic errors were made. This comprehensive verification process helps to ensure the accuracy of our solution and builds confidence in our problem-solving abilities. Now that we have verified the solution, we can confidently present our final answer.
Final Answer
After carefully working through the steps and verifying our solution, we can now state the final answer to the inequality 2(4 + 2x) ≥ 5x + 5. The solution to the inequality is x ≤ 3. This means that all values of 'x' that are less than or equal to 3 will satisfy the original inequality. We arrived at this solution by first distributing the 2 on the left side of the inequality, then combining like terms to isolate the variable 'x'. This process involved algebraic manipulations that preserved the inequality, ensuring that we maintained the correct relationship between the expressions. We then expressed the solution in various forms, including inequality notation (x ≤ 3) and interval notation (-∞, 3]. These notations provide clear and concise ways to represent the solution set, which includes all real numbers less than or equal to 3. To ensure the accuracy of our solution, we performed verification by testing values within and outside the solution set in the original inequality. This step confirmed that our solution is correct and that the values within the solution set satisfy the inequality, while values outside the solution set do not. The final answer, x ≤ 3, provides a complete and accurate solution to the given inequality. It represents a range of values that satisfy the condition stated in the inequality. This solution can be used in various applications, such as determining the range of possible values for a variable in a mathematical model or solving real-world problems involving constraints and limitations. By presenting the final answer clearly and concisely, we have effectively communicated the solution to the problem. This step-by-step approach, from understanding the problem to verifying the solution, is a valuable strategy for solving inequalities and other mathematical problems. With a solid understanding of the concepts and techniques involved, you can confidently tackle a wide range of mathematical challenges.
