Solving Inequalities A Step-by-Step Guide To 3b-7 Less Than 32
In this detailed guide, we will explore the process of solving the inequality 3b - 7 < 32. Inequalities are mathematical expressions that use symbols like '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to) to compare values. Solving an inequality means finding the range of values that satisfy the given condition. This comprehensive guide will walk you through each step, ensuring a clear understanding of how to arrive at the correct solution. This article is designed to help anyone, regardless of their mathematical background, understand the process of solving linear inequalities. We'll break down each step with clear explanations and examples. Let's dive in and master the art of solving inequalities!
Understanding Inequalities
Before we dive into the specific problem, let's clarify what inequalities are and how they differ from equations. An equation is a statement that two expressions are equal, indicated by the '=' sign. For instance, 2x + 3 = 7 is an equation. An inequality, on the other hand, compares two expressions using symbols that indicate a range of values. For example, x < 5 means that x can be any value less than 5, but not equal to 5. The inequality x ≤ 5 means x can be any value less than or equal to 5. Similarly, x > 5 means x is greater than 5, and x ≥ 5 means x is greater than or equal to 5. Understanding these symbols is crucial for interpreting and solving inequalities correctly. Inequalities often have infinitely many solutions, which are usually represented as a range on a number line or in interval notation. This contrasts with equations, which typically have a finite number of solutions. When we solve an inequality, we are essentially finding all the values that make the inequality true. This could be a range of numbers, such as all numbers less than a certain value, or all numbers greater than a certain value. The solution to an inequality is not just a single number, but a set of numbers. For example, the solution to x > 3 is all numbers greater than 3. This set includes 3.1, 4, 5, 10, 100, and so on, up to infinity. This is why understanding the concept of a range of solutions is fundamental to working with inequalities. The solution set can be visualized on a number line, which helps in understanding the range of values that satisfy the inequality. We use different notations to represent the solution set, such as interval notation and set-builder notation. Interval notation uses parentheses and brackets to denote whether the endpoints are included in the solution, while set-builder notation uses curly braces and a variable to define the set of solutions. Mastering these notations is essential for expressing the solutions of inequalities clearly and accurately.
Step-by-Step Solution for 3b - 7 < 32
Now, let's tackle the given inequality: 3b - 7 < 32. Solving this inequality involves isolating the variable b on one side of the inequality sign. We'll follow a step-by-step process similar to solving equations, but with one important difference to keep in mind, which we'll discuss later. The first step in solving this inequality is to isolate the term with the variable. To do this, we need to eliminate the constant term (-7) from the left side of the inequality. We can achieve this by adding 7 to both sides of the inequality. This maintains the balance of the inequality, similar to how we add the same number to both sides of an equation. Adding 7 to both sides, we get: 3b - 7 + 7 < 32 + 7. Simplifying this, we have 3b < 39. This step is crucial because it brings us closer to isolating the variable b. By adding the same number to both sides, we ensure that the inequality remains valid. The next step is to isolate the variable b completely. Currently, b is being multiplied by 3. To undo this multiplication, we need to divide both sides of the inequality by 3. Dividing both sides by 3, we get: (3b)/3 < 39/3. Simplifying this, we have b < 13. This is the solution to the inequality. It means that any value of b that is less than 13 will satisfy the original inequality 3b - 7 < 32. The solution set includes all numbers less than 13, but not 13 itself. We can represent this solution on a number line by drawing an open circle at 13 and shading the line to the left, indicating all values less than 13. In interval notation, the solution is written as (-∞, 13), where the parenthesis indicates that 13 is not included in the solution set. Understanding these representations helps to visualize and communicate the solution effectively.
Step 1: Isolate the Variable Term
To begin, we need to isolate the term containing the variable, which in this case is 3b. We achieve this by adding 7 to both sides of the inequality. Adding the same value to both sides ensures that the inequality remains balanced and the relationship between the two sides is maintained. This is a fundamental principle in solving both equations and inequalities. By adding 7 to both sides, we effectively cancel out the -7 on the left side, moving us closer to isolating b. The equation now transforms as follows: 3b - 7 + 7 < 32 + 7. This step is a crucial foundation for solving the inequality, setting the stage for isolating the variable b and finding its possible values. It's important to perform this step carefully, ensuring that the same operation is applied to both sides of the inequality to maintain its balance. This process is analogous to keeping a scale balanced by adding the same weight to both sides. The result of this addition is a simplified inequality that is easier to work with. It sets up the next step, which involves further isolating the variable to determine the solution set. Mastering this step is key to solving more complex inequalities and understanding the principles behind algebraic manipulations.
Step 2: Simplify the Inequality
After adding 7 to both sides, we simplify the inequality. This involves performing the arithmetic operations to reduce the expression to its simplest form. On the left side, -7 and +7 cancel each other out, leaving us with just 3b. On the right side, 32 + 7 equals 39. So, our inequality now looks like this: 3b < 39. Simplifying the inequality is a crucial step because it makes the problem easier to visualize and solve. By reducing the equation to its simplest form, we eliminate unnecessary clutter and focus on the essential relationship between the variable and the constant. This step not only makes the inequality easier to solve but also reduces the chances of making errors in the subsequent steps. It's like cleaning up your workspace before starting a project – a clean and organized equation is much easier to work with. The simplified inequality 3b < 39 clearly shows that 3b is less than 39. This sets the stage for the next step, which involves isolating the variable b by dividing both sides of the inequality by 3. Simplification is a fundamental skill in algebra, and mastering this step is essential for solving a wide range of mathematical problems. It allows us to break down complex expressions into manageable components, making the overall problem-solving process more efficient and accurate.
Step 3: Isolate the Variable
Now, to completely isolate b, we need to undo the multiplication by 3. We do this by dividing both sides of the inequality by 3. This is a valid operation as long as we divide both sides by the same positive number, which in this case is 3. Dividing both sides by the same positive number maintains the balance of the inequality, similar to how dividing both sides of an equation by the same number preserves the equality. The process looks like this: (3b)/3 < 39/3. When we perform this division, the 3 in the numerator and the 3 in the denominator on the left side cancel each other out, leaving us with just b. On the right side, 39 divided by 3 is 13. So, the inequality becomes: b < 13. This step is crucial because it isolates the variable b, giving us the solution to the inequality. It's important to remember that if we were to divide (or multiply) both sides by a negative number, we would need to flip the inequality sign. However, since we are dividing by a positive number (3), the inequality sign remains the same. Isolating the variable is a fundamental technique in algebra, and it's the key to solving many types of equations and inequalities. Once the variable is isolated, we can clearly see the solution set, which in this case is all values of b that are less than 13.
Step 4: Determine the Solution
After dividing both sides by 3, we arrive at the solution: b < 13. This inequality tells us that the solution set includes all values of b that are strictly less than 13. It's crucial to understand that 13 itself is not included in the solution set because the inequality uses the '<' symbol, which means 'less than', not 'less than or equal to'. The solution b < 13 can be visualized on a number line. To represent this, we draw an open circle at 13 to indicate that 13 is not included, and then we shade the line to the left of 13, representing all the numbers less than 13. This visual representation helps to understand the infinite range of values that satisfy the inequality. In interval notation, the solution is written as (-∞, 13). The parenthesis on both ends indicates that neither negative infinity nor 13 is included in the interval. This notation is a concise way of representing the solution set. The solution b < 13 means that any number less than 13, such as 12, 10, 0, -5, or -100, will make the original inequality 3b - 7 < 32 true. To verify this, you can substitute any of these values into the original inequality and check if the inequality holds. For example, if we substitute b = 10, we get 3(10) - 7 < 32, which simplifies to 23 < 32, which is true. Understanding the solution set is the ultimate goal of solving an inequality. It gives us a clear picture of the range of values that satisfy the given condition. Mastering this step is essential for applying inequalities in real-world problems and for further studies in mathematics.
Choosing the Correct Answer
Now that we have solved the inequality 3b - 7 < 32 and found the solution b < 13, we can confidently choose the correct answer from the given options. Let's revisit the options:
A. b < 8.33 B. b < 5 C. b < 39 D. b < 13
Comparing our solution b < 13 with the options, it is clear that option D is the correct answer. The other options represent different ranges of values, but only option D accurately reflects the solution we derived through our step-by-step process. Option A, b < 8.33, is incorrect because it limits the solution to values less than 8.33, while our solution includes all values less than 13. Option B, b < 5, is also incorrect for the same reason – it's too restrictive. Option C, b < 39, is incorrect because while it includes the correct solution, it also includes values that do not satisfy the original inequality. For instance, if b were 15, which is less than 39, the original inequality 3b - 7 < 32 would become 3(15) - 7 < 32, which simplifies to 38 < 32, which is false. Therefore, option C includes values that are not part of the solution set. Option D, b < 13, is the only option that precisely matches our solution. It includes all values less than 13 and excludes any values that are greater than or equal to 13. This is why it is the correct answer. Choosing the correct answer is the final step in the problem-solving process. It demonstrates that you have not only understood the steps involved in solving the inequality but also that you can accurately interpret the solution and match it with the appropriate representation.
Key Concepts Recap
To summarize, let's recap the key concepts we've covered in this guide. Understanding these concepts is crucial for mastering inequalities and tackling more complex mathematical problems. The first key concept is understanding inequality symbols. Knowing the difference between '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to) is fundamental. These symbols dictate the range of values that satisfy the inequality. The second key concept is isolating the variable. This involves performing algebraic operations to get the variable by itself on one side of the inequality. We do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. The third key concept is the golden rule of inequalities: When multiplying or dividing both sides of an inequality by a negative number, you must flip the inequality sign. This is a crucial rule to remember, as forgetting to flip the sign will lead to an incorrect solution. The fourth key concept is simplifying the inequality. This involves performing arithmetic operations to reduce the inequality to its simplest form, making it easier to solve. Simplifying the inequality helps to clarify the relationship between the variable and the constant terms. The fifth key concept is representing the solution. The solution to an inequality can be represented in several ways, including on a number line and in interval notation. A number line provides a visual representation of the solution set, while interval notation is a concise way of expressing the range of values. The sixth key concept is verifying the solution. To ensure that your solution is correct, you can substitute values from the solution set back into the original inequality to check if they satisfy the inequality. This is a good practice to avoid errors and build confidence in your problem-solving skills. By mastering these key concepts, you'll be well-equipped to solve a wide range of inequalities and apply them in various mathematical contexts. Inequalities are not just abstract mathematical concepts; they have practical applications in various fields, including economics, engineering, and computer science.
Conclusion
In conclusion, we have thoroughly explored the process of solving the inequality 3b - 7 < 32. By following a step-by-step approach, we successfully isolated the variable b and determined the solution to be b < 13. This means that any value of b less than 13 will satisfy the original inequality. We also discussed the importance of understanding inequality symbols, isolating the variable, simplifying the inequality, and representing the solution in different forms, such as on a number line and in interval notation. Furthermore, we highlighted the crucial rule of flipping the inequality sign when multiplying or dividing by a negative number. By mastering these concepts, you can confidently tackle a wide range of inequality problems. Inequalities are a fundamental part of mathematics and have numerous applications in real-world scenarios. From determining the feasible region in linear programming to understanding constraints in optimization problems, inequalities play a vital role in various fields. The ability to solve inequalities is not just a mathematical skill; it's a valuable tool for problem-solving and decision-making in many aspects of life. As you continue your mathematical journey, remember to practice regularly and apply these concepts to different problems. The more you practice, the more comfortable and confident you will become in solving inequalities and other mathematical challenges. Keep exploring, keep learning, and keep solving!
