Solving Inequalities A Step-by-Step Guide To 8x + 6 < 54
In the realm of mathematics, inequalities play a crucial role in defining relationships between quantities that are not necessarily equal. Understanding how to solve inequalities is a fundamental skill that extends far beyond the classroom, finding applications in various fields like economics, engineering, and computer science. This comprehensive guide will delve into the process of solving the inequality 8x + 6 < 54, providing a step-by-step approach along with explanations to enhance your understanding. This exploration will not only provide a solution to the given inequality but also equip you with the skills to tackle similar problems with confidence.
Understanding Inequalities
Before we dive into the solution, let's first grasp the concept of inequalities. Unlike equations that use an equals sign (=) to show that two expressions are equivalent, inequalities use symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to express relationships where two quantities are not necessarily equal. The inequality 8x + 6 < 54 states that the expression 8x + 6 is less than 54. Solving this inequality involves finding the range of values for x that satisfy this condition. These values, unlike in equations where we often find a single solution, will form a set of numbers, typically an interval on the number line. Understanding this difference between equations and inequalities is crucial for approaching problem-solving effectively. Think of inequalities as providing a broader range of possible solutions, reflecting real-world scenarios where variables can fall within certain limits rather than being confined to a single value.
Step-by-Step Solution
Now, let's embark on the journey of solving the inequality 8x + 6 < 54. We will follow a series of algebraic steps to isolate the variable x and determine its possible values. Each step is designed to maintain the integrity of the inequality while simplifying the expression. Our goal is to gradually peel away the layers surrounding x until we have a clear picture of its permissible range. This process mirrors the logic used in solving equations, with a few key differences that we will highlight along the way.
1. Isolate the term with x:
The first step is to isolate the term containing x, which in this case is 8x. To do this, we need to eliminate the constant term, which is +6. We achieve this by subtracting 6 from both sides of the inequality. This maintains the balance of the inequality, ensuring that the relationship between the two sides remains intact. Subtracting 6 from both sides gives us:
8x + 6 - 6 < 54 - 6
This simplifies to:
8x < 48
This step is crucial because it brings us closer to isolating x. By removing the constant term, we create a simpler inequality that is easier to manipulate. It's like clearing away the underbrush in a forest to get a better view of the trees.
2. Solve for x:
Now that we have 8x < 48, the next step is to isolate x completely. Since x is being multiplied by 8, we need to perform the inverse operation, which is division. We will divide both sides of the inequality by 8. Again, this is a fundamental rule of algebra: what you do to one side, you must do to the other to maintain balance. Dividing both sides by 8 gives us:
8x / 8 < 48 / 8
This simplifies to:
x < 6
This is our solution! It tells us that x can be any value less than 6. This is a crucial point to understand: the solution is not a single number, but a range of numbers. Any number less than 6 will satisfy the original inequality. This understanding is key to interpreting the solution correctly and applying it in different contexts.
Representing the Solution
There are several ways to represent the solution x < 6. We can use interval notation, a number line, or simply state it in words. Each method offers a different perspective on the solution set.
1. Interval Notation:
Interval notation is a concise way to represent a range of numbers. For x < 6, the interval notation is (-∞, 6). The parenthesis indicates that 6 is not included in the solution set, as the inequality is strictly less than. The -∞ symbol represents negative infinity, indicating that the solution extends indefinitely in the negative direction. This notation is particularly useful for expressing complex solution sets, making it a standard tool in mathematical communication.
2. Number Line:
A number line provides a visual representation of the solution. To represent x < 6 on a number line, we draw a line and mark the point 6. We then draw an open circle at 6 to indicate that it is not included in the solution. Finally, we shade the line to the left of 6, representing all numbers less than 6. This visual representation can be particularly helpful for students who are just beginning to understand inequalities, as it provides a concrete way to see the range of possible solutions.
3. Words:
We can also simply state the solution in words: "x is less than 6." While this method is straightforward, it may not be as precise or concise as interval notation or the number line representation, especially when dealing with more complex inequalities.
Special Cases and Considerations
While the process of solving 8x + 6 < 54 is relatively straightforward, there are a few special cases and considerations to keep in mind when dealing with inequalities in general. These nuances are crucial for avoiding common errors and ensuring accurate solutions.
1. Dividing or Multiplying by a Negative Number:
One of the most important rules to remember is that when you divide or multiply both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if we had -2x < 4, dividing both sides by -2 would give us x > -2 (note the flipped inequality sign). This rule stems from the nature of negative numbers and their effect on the order of values on the number line. Failing to flip the sign is a common mistake that can lead to incorrect solutions.
2. Compound Inequalities:
Compound inequalities involve two or more inequalities combined into a single statement. For example, -3 < x ≤ 5 is a compound inequality that states that x is greater than -3 and less than or equal to 5. Solving compound inequalities involves addressing each part of the inequality separately while maintaining the overall relationship. These inequalities often require careful attention to detail to ensure that the solution set is accurately determined.
3. No Solution or Infinite Solutions:
In some cases, inequalities may have no solution or infinite solutions. An inequality with no solution is one where there is no value of the variable that can satisfy the condition. For example, x + 1 < x has no solution. An inequality with infinite solutions is one where any value of the variable will satisfy the condition. For example, x + 1 > x is true for all real numbers. Recognizing these scenarios is important for correctly interpreting the results of your problem-solving efforts.
Real-World Applications
The ability to solve inequalities is not just an academic exercise; it has numerous practical applications in real-world scenarios. Inequalities are used to model situations where quantities have constraints or limitations. They provide a powerful tool for making decisions and solving problems in various fields.
1. Budgeting and Finance:
Inequalities can be used to represent budget constraints. For example, if you have a budget of $100 for groceries, you can use an inequality to represent the total amount you can spend. If x represents the amount spent on fruits and vegetables, and y represents the amount spent on other groceries, the inequality x + y ≤ 100 would represent your budget constraint. This allows you to explore different spending scenarios while staying within your budget.
2. Engineering and Physics:
Inequalities are used in engineering to define safety margins and tolerances. For example, the maximum load a bridge can support can be represented by an inequality. Similarly, in physics, inequalities can be used to describe the range of possible values for physical quantities, such as temperature or pressure. These applications highlight the critical role of inequalities in ensuring safety and reliability in various systems.
3. Computer Science:
In computer science, inequalities are used in algorithms and optimization problems. For example, inequalities can be used to define constraints in linear programming problems, which are used to optimize resource allocation. They are also used in the analysis of algorithms to determine their efficiency and performance. This demonstrates the importance of inequalities in the development of efficient and effective computational solutions.
Practice Problems
To solidify your understanding of solving inequalities, let's work through a few practice problems. These problems will allow you to apply the concepts and techniques we have discussed and build your confidence in tackling more complex inequalities.
1. Solve the inequality 3x - 5 > 10:
- Add 5 to both sides: 3x > 15
- Divide both sides by 3: x > 5
- Solution in interval notation: (5, ∞)
2. Solve the inequality -2x + 4 ≤ 8:
- Subtract 4 from both sides: -2x ≤ 4
- Divide both sides by -2 (and flip the inequality sign): x ≥ -2
- Solution in interval notation: [-2, ∞)
3. Solve the compound inequality 1 < 2x + 3 < 7:
- Subtract 3 from all parts: -2 < 2x < 4
- Divide all parts by 2: -1 < x < 2
- Solution in interval notation: (-1, 2)
By working through these practice problems, you can reinforce your understanding of the steps involved in solving inequalities and develop your problem-solving skills.
Conclusion
Solving inequalities is a fundamental skill in mathematics with wide-ranging applications. By understanding the basic principles and techniques, you can confidently tackle a variety of inequality problems. In this guide, we have explored the step-by-step process of solving the inequality 8x + 6 < 54, along with various methods for representing the solution. We have also discussed special cases and considerations, as well as real-world applications of inequalities. Remember to practice regularly and apply these concepts to different problems to further enhance your understanding. With dedication and practice, you can master the art of solving inequalities and unlock their power in various fields of study and application.
