Solving Inequalities A Step By Step Guide To Solve For X In 3x + 3 - X + (-7) > 6
In the realm of mathematics, solving for variables is a fundamental skill. This article delves into the process of solving a linear inequality, specifically the inequality 3x + 3 - x + (-7) > 6. We will break down the steps involved, providing a clear and concise explanation for each stage. Linear inequalities, unlike linear equations, involve comparing two expressions using inequality symbols such as '>', '<', '≥', or '≤'. Solving them requires a slightly different approach, but the underlying principles of algebraic manipulation remain the same. Understanding how to solve linear inequalities is crucial for various mathematical applications, including optimization problems, graphing inequalities, and determining solution sets.
To effectively solve the inequality 3x + 3 - x + (-7) > 6, we will follow a series of steps designed to isolate the variable 'x' on one side of the inequality. This involves combining like terms, simplifying the expression, and applying inverse operations. The ultimate goal is to determine the range of values for 'x' that satisfy the given inequality. This solution set will represent all the possible values of 'x' that make the inequality true. Let's embark on this journey of algebraic manipulation and unravel the solution to this inequality, step by step.
Step 1: Simplify the Inequality
The first step in solving the inequality is to simplify both sides by combining like terms. In this case, we have terms involving 'x' and constant terms. Let's group them together:
3x + 3 - x + (-7) > 6
Combine the 'x' terms: 3x - x = 2x
Combine the constant terms: 3 + (-7) = -4
Now, the inequality becomes:
2x - 4 > 6
This simplified form is easier to work with and allows us to proceed with isolating the variable 'x'. By combining like terms, we reduce the complexity of the inequality and bring it closer to a form where we can apply inverse operations to solve for 'x'. This simplification process is a crucial step in solving any algebraic equation or inequality, as it streamlines the expression and makes it more manageable.
Step 2: Isolate the Variable Term
Our next goal is to isolate the term containing 'x' (which is 2x) on one side of the inequality. To do this, we need to eliminate the constant term (-4) on the left side. We can achieve this by adding 4 to both sides of the inequality. Remember, when working with inequalities, it's crucial to perform the same operation on both sides to maintain the balance and ensure the inequality remains valid.
Adding 4 to both sides:
2x - 4 + 4 > 6 + 4
This simplifies to:
2x > 10
Now, the term with 'x' is isolated on the left side, making it easier to solve for 'x'. This step is crucial because it brings us closer to our ultimate goal of determining the value or range of values for 'x' that satisfy the inequality. By isolating the variable term, we pave the way for the final step of dividing to solve for 'x'.
Step 3: Solve for x
To finally solve for 'x', we need to get 'x' by itself. Since 'x' is being multiplied by 2, we can undo this operation by dividing both sides of the inequality by 2. Again, it's essential to perform the same operation on both sides to maintain the integrity of the inequality. Also, remember that when dividing (or multiplying) both sides of an inequality by a negative number, you must flip the inequality sign. However, in this case, we are dividing by a positive number (2), so we don't need to worry about flipping the sign.
Dividing both sides by 2:
(2x) / 2 > 10 / 2
This simplifies to:
x > 5
Therefore, the solution to the inequality 3x + 3 - x + (-7) > 6 is x > 5. This means that any value of 'x' greater than 5 will satisfy the original inequality. We have successfully isolated 'x' and determined the solution set for the given inequality. This final step completes the process of solving for 'x' and provides us with the answer to the problem.
Understanding the Solution: x > 5
The solution x > 5 represents a range of values for 'x' that satisfy the original inequality. It means that any number greater than 5 will make the inequality true. For example, if we substitute x = 6 into the original inequality, we get:
3(6) + 3 - 6 + (-7) > 6
18 + 3 - 6 - 7 > 6
8 > 6 (which is true)
However, if we substitute a value less than or equal to 5, such as x = 5, we get:
3(5) + 3 - 5 + (-7) > 6
15 + 3 - 5 - 7 > 6
6 > 6 (which is false)
This confirms that the solution x > 5 is correct. The inequality holds true for all values of 'x' greater than 5, but not for values less than or equal to 5. This understanding of the solution set is crucial for interpreting the results and applying them in various contexts.
Solution Set and Graphical Representation
The solution set for the inequality x > 5 includes all real numbers greater than 5. This can be represented graphically on a number line. We draw an open circle at 5 to indicate that 5 is not included in the solution set, and then we shade the region to the right of 5 to represent all values greater than 5. This graphical representation provides a visual understanding of the solution set and helps to illustrate the range of values that satisfy the inequality.
The solution set can also be expressed in interval notation as (5, ∞). The parenthesis indicates that 5 is not included in the interval, and the infinity symbol represents that the interval extends indefinitely to the right. This interval notation is a concise way to represent the solution set and is commonly used in higher-level mathematics.
Choosing the Correct Answer
Based on our step-by-step solution, we have determined that x > 5 is the solution to the inequality 3x + 3 - x + (-7) > 6. Therefore, the correct answer from the given options is:
B. x > 5
We have successfully solved the inequality and identified the correct answer choice. This demonstrates the importance of following a systematic approach when solving mathematical problems, ensuring accuracy and clarity in each step.
Key Concepts and Takeaways
This exercise highlights several key concepts in solving linear inequalities:
- Combining like terms: Simplifying the inequality by grouping terms with 'x' and constant terms.
- Isolating the variable term: Using inverse operations to isolate the term containing 'x' on one side of the inequality.
- Solving for x: Dividing both sides of the inequality by the coefficient of 'x' to find the solution.
- Understanding the solution set: Recognizing that the solution represents a range of values for 'x' that satisfy the inequality.
- Graphical representation: Visualizing the solution set on a number line.
By mastering these concepts, you can confidently solve a wide range of linear inequalities. Remember to always follow the steps carefully, perform the same operations on both sides of the inequality, and pay attention to the direction of the inequality sign.
Applications of Linear Inequalities
Linear inequalities have numerous applications in real-world scenarios. They are used to model situations where there is a constraint or a range of acceptable values. Here are a few examples:
- Budgeting: Determining the maximum amount of money you can spend on different items while staying within your budget.
- Optimization: Finding the maximum or minimum value of a function subject to certain constraints.
- Physics: Describing the range of possible values for physical quantities such as speed, distance, or time.
- Statistics: Defining confidence intervals and hypothesis testing.
Understanding linear inequalities is essential for solving problems in various fields. By mastering the techniques discussed in this article, you can apply them to real-world situations and make informed decisions.
Practice Problems
To further solidify your understanding of solving linear inequalities, try solving the following practice problems:
- Solve for x: 2x - 5 < 3
- Solve for x: -4x + 7 ≥ 15
- Solve for x: 5x + 2 > 3x - 4
By working through these problems, you will gain more experience and confidence in solving linear inequalities. Remember to follow the steps outlined in this article and check your answers to ensure accuracy.
Conclusion
In this comprehensive guide, we have explored the process of solving the linear inequality 3x + 3 - x + (-7) > 6. We have broken down the steps involved, providing a clear explanation for each stage. From simplifying the inequality to isolating the variable term and finally solving for 'x', we have demonstrated a systematic approach to solving linear inequalities.
By understanding the key concepts and techniques discussed in this article, you can confidently solve a wide range of linear inequalities. Remember to practice regularly and apply these skills to real-world scenarios. With dedication and effort, you can master the art of solving linear inequalities and excel in your mathematical journey.
