Solving 3-4x ≥ X-7 A Step-by-Step Guide
Solving inequalities is a fundamental concept in mathematics, and mastering it is essential for various applications in algebra, calculus, and beyond. In this comprehensive guide, we will delve into the process of solving the inequality 3-4x ≥ x-7, providing a step-by-step explanation and offering valuable insights into the underlying principles. This guide is tailored for students, educators, and anyone seeking to enhance their understanding of mathematical inequalities. We will cover the key steps involved in isolating the variable, interpreting the solution, and representing it graphically. By the end of this guide, you will be well-equipped to tackle similar inequalities with confidence and precision. Whether you are a beginner or an experienced learner, this resource will serve as a valuable tool in your mathematical journey.
Understanding Inequalities
Before we dive into the specifics of the inequality 3-4x ≥ x-7, let's establish a solid understanding of what inequalities are and how they differ from equations. Inequalities are mathematical statements that compare two expressions using symbols such as >, <, ≥, and ≤. These symbols indicate that the two expressions are not necessarily equal, but rather one is greater than, less than, greater than or equal to, or less than or equal to the other. This distinction is crucial because it affects how we solve and interpret these mathematical statements. Unlike equations, which typically have a single solution or a finite set of solutions, inequalities often have a range of solutions. For instance, the inequality x > 3 means that any number greater than 3 satisfies the condition, leading to an infinite number of solutions. Understanding the nature of these solutions is essential for accurately representing them, whether on a number line or in interval notation. Moreover, certain operations, such as multiplying or dividing by a negative number, require special attention when dealing with inequalities. These operations necessitate flipping the inequality sign to maintain the truth of the statement. Grasping these nuances is vital for avoiding common pitfalls and ensuring the correctness of your solutions. In the following sections, we will explore these concepts in greater detail, providing examples and practical tips to solidify your understanding of inequalities.
Step-by-Step Solution of 3-4x ≥ x-7
Now, let's break down the solution of the inequality 3-4x ≥ x-7 into manageable steps. Our primary goal is to isolate the variable 'x' on one side of the inequality. This process involves applying algebraic operations to both sides of the inequality while maintaining its balance. The first step is to gather all terms containing 'x' on one side and constant terms on the other. To achieve this, we can add 4x to both sides of the inequality, which yields 3 ≥ 5x - 7. This move eliminates the '-4x' term from the left side, simplifying the expression. Next, we need to isolate the term with 'x' further. We can do this by adding 7 to both sides of the inequality. This results in 10 ≥ 5x. Now, we have all the 'x' terms on one side and the constant terms on the other, making it easier to isolate 'x' completely. The final step in isolating 'x' is to divide both sides of the inequality by 5. This gives us 2 ≥ x, which can also be written as x ≤ 2. This is our solution: 'x' is less than or equal to 2. It's crucial to remember that when dividing or multiplying an inequality by a negative number, we need to flip the direction of the inequality sign. However, in this case, we divided by a positive number, so the inequality sign remains the same. This step-by-step approach ensures clarity and accuracy in solving inequalities. In the next sections, we will discuss how to interpret and represent this solution graphically.
Step 1: Combine Like Terms
The initial step in solving the inequality 3-4x ≥ x-7 involves combining like terms to simplify the expression. This is a fundamental algebraic technique that helps streamline the equation and make it easier to isolate the variable. Our objective here is to gather all terms containing 'x' on one side of the inequality and all constant terms on the other side. This process sets the stage for isolating 'x' and determining its possible values. To begin, we observe that the inequality has '-4x' on the left side and '+x' on the right side. To consolidate these terms, we can add 4x to both sides of the inequality. This operation eliminates the '-4x' term from the left side, resulting in a simpler expression. Adding 4x to both sides ensures that the inequality remains balanced, maintaining the integrity of the mathematical statement. The resulting inequality is 3 ≥ x + 4x - 7, which simplifies to 3 ≥ 5x - 7. Now, all the 'x' terms are on the right side, and we can proceed to gather the constant terms. The next step is to move the constant term '-7' from the right side to the left side. We can achieve this by adding 7 to both sides of the inequality. This operation isolates the 'x' term further, bringing us closer to the final solution. Adding 7 to both sides gives us 3 + 7 ≥ 5x, which simplifies to 10 ≥ 5x. At this point, we have successfully combined like terms, placing all 'x' terms on one side and all constant terms on the other. This simplification is crucial for the subsequent steps in solving the inequality. In the following sections, we will continue to isolate 'x' and determine the range of values that satisfy the inequality.
Step 2: Isolate the Variable
After combining like terms, the next crucial step is to isolate the variable 'x'. This involves performing algebraic operations to get 'x' by itself on one side of the inequality. We have reached the simplified form 10 ≥ 5x. To isolate 'x', we need to eliminate the coefficient 5 that is multiplying 'x'. This can be achieved by dividing both sides of the inequality by 5. Dividing both sides by the same positive number ensures that the inequality remains balanced and the solution set is not altered. When we divide both sides by 5, we get 10/5 ≥ 5x/5, which simplifies to 2 ≥ x. This inequality can also be written as x ≤ 2, which is a more intuitive way to express the solution. The inequality x ≤ 2 means that 'x' can be any number that is less than or equal to 2. This includes numbers such as 2, 1, 0, -1, -2, and so on. It's important to note that when dividing or multiplying an inequality by a negative number, the direction of the inequality sign must be reversed. However, in this case, we divided by a positive number (5), so the inequality sign remains the same. Isolating the variable is a critical step in solving inequalities, as it allows us to clearly identify the range of values that satisfy the original inequality. In this case, we have successfully isolated 'x' and found that it must be less than or equal to 2. In the next section, we will discuss how to interpret this solution and represent it graphically.
Interpreting the Solution
Interpreting the solution to an inequality is a vital step in the problem-solving process. Once we have isolated the variable, we need to understand what the solution actually means in the context of the original problem. In our case, the solution to the inequality 3-4x ≥ x-7 is x ≤ 2. This means that any value of 'x' that is less than or equal to 2 will satisfy the original inequality. To fully grasp this, let's consider a few examples. If we substitute x = 2 into the original inequality, we get 3 - 4(2) ≥ 2 - 7, which simplifies to 3 - 8 ≥ -5, and further to -5 ≥ -5. This is a true statement, confirming that x = 2 is indeed a solution. Now, let's try a value less than 2, such as x = 0. Substituting this into the original inequality gives us 3 - 4(0) ≥ 0 - 7, which simplifies to 3 ≥ -7. This is also a true statement, indicating that x = 0 is a solution. On the other hand, if we try a value greater than 2, such as x = 3, we get 3 - 4(3) ≥ 3 - 7, which simplifies to 3 - 12 ≥ -4, and further to -9 ≥ -4. This is a false statement, demonstrating that x = 3 is not a solution. These examples illustrate that the solution x ≤ 2 represents a range of values, specifically all numbers less than or equal to 2. This is a key difference between inequalities and equations, which typically have a single solution or a finite set of solutions. Understanding the solution set is crucial for various applications, such as graphing the solution on a number line or using it in further calculations. In the next section, we will explore how to represent this solution graphically.
Representing the Solution Graphically
Graphically representing the solution to an inequality provides a visual understanding of the range of values that satisfy the inequality. For the solution x ≤ 2, we can use a number line to illustrate all the values of 'x' that are less than or equal to 2. A number line is a simple yet powerful tool for visualizing inequalities. It consists of a horizontal line with numbers marked at equal intervals. To represent x ≤ 2, we first locate the number 2 on the number line. Since the inequality includes
