Solve The Inequality -4 < 2 - 2x ≤ 14. Please Provide A Step-by-step Solution.
In the realm of mathematics, inequalities play a crucial role in defining ranges and constraints. Mastering the art of solving inequalities is fundamental for various mathematical disciplines, from basic algebra to advanced calculus. This article delves into a specific inequality, -4 < 2 - 2x ≤ 14, providing a comprehensive, step-by-step guide to its solution. We will explore the underlying concepts, unravel the solving techniques, and equip you with the skills to confidently tackle similar mathematical challenges. Whether you're a student honing your algebraic prowess or simply seeking a refresher on inequality solutions, this guide is your roadmap to success. By the end of this journey, you'll not only understand how to solve this particular inequality but also grasp the broader principles that govern inequality manipulations.
Understanding Inequalities
Before we dive into solving the specific inequality, let's establish a solid foundation by understanding the fundamental concepts of inequalities. Unlike equations, which assert the equality of two expressions, inequalities express a relationship of greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). Inequalities define a range of values that satisfy a given condition, rather than a single solution.
The inequality -4 < 2 - 2x ≤ 14 is a compound inequality, meaning it combines two inequalities into a single statement. It essentially states that the expression 2 - 2x is both greater than -4 and less than or equal to 14. This type of inequality is often solved by treating it as two separate inequalities and then combining the results. Understanding the 'and' condition is crucial here; a solution must satisfy both inequalities simultaneously.
To further illustrate this concept, consider a number line. The solution to an inequality is represented by a region on the number line, which can be an open interval (values between two points, not including the points themselves), a closed interval (values between two points, including the points themselves), or a combination of both. The symbols '<' and '>' correspond to open intervals, while '≤' and '≥' correspond to closed intervals. In our case, '-4 < 2 - 2x' represents an open interval on one side, and '2 - 2x ≤ 14' represents a closed interval on the other, leading to a solution that is bounded on both ends.
Isolating the Variable: The Core Strategy
The primary goal in solving any inequality is to isolate the variable on one side of the inequality sign. This process involves performing a series of algebraic operations, ensuring that the variable stands alone, revealing the range of values that satisfy the inequality. The techniques used for isolating the variable in inequalities are similar to those used in solving equations, with one critical distinction: multiplying or dividing by a negative number reverses the direction of the inequality sign. This rule is paramount to maintaining the accuracy of the solution.
To effectively isolate the variable, we often employ the following strategies:
- Addition and Subtraction: Adding or subtracting the same value from all parts of the inequality does not change the direction of the inequality signs. This allows us to eliminate constant terms from the side containing the variable.
- Multiplication and Division: Multiplying or dividing all parts of the inequality by a positive value does not change the direction of the inequality signs. However, multiplying or dividing by a negative value does require reversing the inequality signs.
- Distribution: If the inequality involves parentheses, we may need to distribute a value across the terms within the parentheses before proceeding with other operations.
- Combining Like Terms: Simplifying each side of the inequality by combining like terms can make the subsequent steps easier.
In the context of our compound inequality, we will apply these techniques systematically to isolate 'x' in the middle, revealing the solution set that satisfies the given conditions. Let's proceed with the first steps in solving the inequality.
Step-by-Step Solution of -4 < 2 - 2x ≤ 14
Now, let's embark on the journey of solving the compound inequality -4 < 2 - 2x ≤ 14 step-by-step. Remember, our ultimate goal is to isolate 'x' in the middle, thereby revealing the range of values that satisfy the inequality. We will approach this by performing the same operations on all three parts of the inequality, maintaining the balance and ensuring the integrity of the solution.
Step 1: Subtract 2 from all parts of the inequality
To begin isolating 'x', we need to eliminate the constant term '2' from the middle part. We achieve this by subtracting 2 from all three sections of the inequality:
-4 - 2 < 2 - 2x - 2 ≤ 14 - 2
This simplifies to:
-6 < -2x ≤ 12
By performing the subtraction, we've successfully moved the constant term to the outer parts of the inequality, bringing us closer to isolating 'x'. The next step involves dealing with the coefficient of 'x'.
Step 2: Divide all parts of the inequality by -2
Here comes the crucial step where we must remember the golden rule of inequalities: dividing or multiplying by a negative number reverses the direction of the inequality signs. We need to divide all parts of the inequality by -2 to isolate 'x'.
(-6) / -2 > (-2x) / -2 ≥ (12) / -2
Notice that we've reversed both inequality signs. Simplifying the expression, we get:
3 > x ≥ -6
Now, let's rewrite this inequality in a more conventional form, where the smaller value is on the left and the larger value is on the right. This simply involves flipping the entire inequality:
-6 ≤ x < 3
This is our solution! It states that 'x' is greater than or equal to -6 and less than 3.
Step 3: Express the solution in interval notation
To provide a concise and standardized representation of the solution, we often use interval notation. Interval notation uses brackets and parentheses to indicate the range of values included in the solution. A square bracket '[ ]' indicates that the endpoint is included in the solution (≤ or ≥), while a parenthesis '( )' indicates that the endpoint is not included (< or >).
In our case, the solution -6 ≤ x < 3 can be expressed in interval notation as:
[-6, 3)
The square bracket on the left indicates that -6 is included in the solution, and the parenthesis on the right indicates that 3 is not included.
Step 4: Represent the solution on a number line
A visual representation of the solution can further enhance understanding. We can depict the solution on a number line by drawing a closed circle at -6 (indicating inclusion) and an open circle at 3 (indicating exclusion), and shading the region between these two points. This shaded region represents all the values of 'x' that satisfy the inequality.
By completing these steps, we have successfully solved the inequality -4 < 2 - 2x ≤ 14. We have isolated the variable, expressed the solution in both inequality and interval notation, and visualized it on a number line. This comprehensive approach solidifies our understanding of the solution and the underlying principles of solving inequalities. In the next section, we will delve deeper into common mistakes to avoid and strategies for verifying your solutions.
Common Mistakes and How to Avoid Them
Solving inequalities, like any mathematical endeavor, is prone to certain common errors. Being aware of these pitfalls and developing strategies to avoid them is crucial for achieving accurate solutions. Let's explore some frequent mistakes and how to navigate them successfully.
1. Forgetting to Reverse the Inequality Sign:
This is perhaps the most common mistake when dealing with inequalities. Remember the cardinal rule: When multiplying or dividing both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. Failing to do so will lead to an incorrect solution set.
How to avoid it: Always double-check your work when multiplying or dividing by a negative number. A helpful technique is to write down the phrase "Reverse the sign!" as a reminder when you perform this operation.
2. Incorrectly Distributing Negative Signs:
When an inequality involves parentheses with a negative sign preceding them, it's essential to distribute the negative sign correctly to all terms inside the parentheses. A missed negative sign can alter the entire equation and lead to a wrong answer.
How to avoid it: Take your time when distributing negative signs. It can be helpful to write out the distribution step explicitly to avoid errors.
3. Mixing Up Inequality Symbols:
It's easy to mix up the greater than (>) and less than (<) symbols, or the greater than or equal to (≥) and less than or equal to (≤) symbols, especially when working quickly. This can lead to a flipped solution set.
How to avoid it: Pay close attention to the direction of the inequality symbols. If you find yourself getting confused, try visualizing the number line. Numbers to the right are greater, and numbers to the left are less.
4. Incorrectly Applying Interval Notation:
Interval notation provides a concise way to represent the solution set, but it's crucial to use the correct notation. Square brackets indicate inclusion of the endpoint, while parentheses indicate exclusion.
How to avoid it: Refer to the definitions of interval notation. If the inequality includes ≤ or ≥, use a square bracket. If the inequality includes < or >, use a parenthesis. When in doubt, visualize the number line representation of the solution.
5. Not Checking the Solution:
One of the best ways to catch errors is to check your solution. Choose a value within your solution set and substitute it back into the original inequality. If the inequality holds true, your solution is likely correct. Similarly, choose a value outside your solution set and verify that it does not satisfy the original inequality.
How to avoid it: Make it a habit to check your solutions, especially on tests and exams. This simple step can save you from making careless mistakes.
By being mindful of these common errors and adopting the suggested strategies, you can significantly improve your accuracy and confidence in solving inequalities. In the final section, we'll provide a comprehensive summary of the key concepts and techniques covered in this article.
Conclusion: Mastering Inequalities
Throughout this comprehensive guide, we have explored the intricacies of solving the inequality -4 < 2 - 2x ≤ 14, delving into the fundamental concepts, step-by-step solution process, and common pitfalls to avoid. By understanding the principles that govern inequalities and mastering the techniques for manipulating them, you are now equipped to tackle a wide range of mathematical challenges.
We began by laying the groundwork, understanding the nature of inequalities as expressions that define a range of values rather than a single solution. We emphasized the importance of isolating the variable, the core strategy for solving inequalities, and highlighted the crucial rule of reversing the inequality sign when multiplying or dividing by a negative number.
We then meticulously walked through the solution process, demonstrating each step with clarity and precision: subtracting constants, dividing by coefficients, and expressing the solution in both inequality and interval notation. We also emphasized the value of visualizing the solution on a number line, providing a concrete representation of the range of values that satisfy the inequality.
Furthermore, we addressed the common mistakes that often plague those new to solving inequalities, such as forgetting to reverse the sign, incorrectly distributing negative signs, and misinterpreting interval notation. By providing strategies to avoid these pitfalls, we empower you to approach inequalities with greater confidence and accuracy.
In essence, solving inequalities is not merely about finding a numerical answer; it's about understanding the relationships between quantities and the constraints that govern them. The skills you've acquired in this guide extend far beyond this specific example, providing a foundation for more advanced mathematical concepts and real-world problem-solving.
As you continue your mathematical journey, remember to practice consistently, seek out challenging problems, and never hesitate to revisit the fundamentals. With dedication and perseverance, you can master the art of solving inequalities and unlock a deeper understanding of the mathematical world around you. Whether you're a student striving for academic excellence or simply a curious mind seeking to expand your knowledge, the principles and techniques presented in this article will serve as invaluable tools in your mathematical toolkit. So, embrace the challenge, continue to explore, and celebrate the power of mathematical thinking!