Solving 3b - 7 < 32 A Step-by-Step Guide To Inequalities

Linear inequalities are mathematical statements that compare two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving inequalities involves finding the range of values that satisfy the given inequality. In this article, we will delve into the process of solving the inequality $3b - 7 < 32$, providing a step-by-step guide and highlighting key concepts.

The given inequality is a linear inequality because the variable 'b' is raised to the power of 1. To solve this inequality, our goal is to isolate the variable 'b' on one side of the inequality sign. We can achieve this by performing algebraic operations on both sides of the inequality, ensuring that we maintain the inequality's balance. The fundamental principle behind solving inequalities is that we can add or subtract the same value from both sides without changing the inequality. Similarly, we can multiply or divide both sides by the same positive value without affecting the inequality. However, if we multiply or divide by a negative value, we must reverse the direction of the inequality sign. Understanding these principles is vital for correctly solving inequalities and interpreting the solutions.

Let's break down the solution into manageable steps:

1. Isolate the term with the variable: Our first step is to isolate the term containing the variable 'b', which is $3b$. To do this, we need to eliminate the constant term, $-7$, from the left side of the inequality. We can accomplish this by adding 7 to both sides of the inequality. Adding the same value to both sides maintains the balance of the inequality and helps us move closer to isolating 'b'. The process of isolating variables is fundamental in solving algebraic equations and inequalities, as it allows us to determine the specific values or range of values that satisfy the given condition. The principle of maintaining balance is crucial in all algebraic manipulations, ensuring that the solution remains accurate and consistent with the initial equation or inequality.

   $3b - 7 + 7 < 32 + 7$

   This simplifies to:

   $3b < 39$

2. Isolate the variable: Now that we have $3b < 39$, we need to isolate 'b' completely. This means we need to get rid of the coefficient 3 that is multiplying 'b'. To do this, we divide both sides of the inequality by 3. Dividing both sides by a positive number does not change the direction of the inequality sign. The coefficient of a variable plays a significant role in determining its value in an equation or inequality. By dividing by the coefficient, we are essentially scaling down the variable to its unit value, which makes it easier to determine its range or specific value. This step is critical in the solution process, as it directly leads to the isolation of the variable and the final solution.

   $\frac{3b}{3} < \frac{39}{3}$

   This simplifies to:

   $b < 13$

The solution to the inequality $3b - 7 < 32$ is $b < 13$. This means that any value of 'b' that is less than 13 will satisfy the original inequality. It's important to understand that this solution represents a range of values, not just a single value. This range includes all real numbers less than 13, extending infinitely towards the negative side. We can visualize this solution on a number line by drawing an open circle at 13 and shading the region to the left, indicating that all values less than 13 are included in the solution set. The open circle signifies that 13 itself is not included in the solution because the inequality is strictly less than, not less than or equal to. Understanding the graphical representation of solutions is crucial in comprehending the extent of the solution set and its implications. It provides a visual aid that complements the algebraic solution, enhancing our understanding of the inequality.

To ensure our solution is correct, we can test a value within the solution range and a value outside the solution range in the original inequality.

1. Test a value within the solution range: Let's choose $b = 10$, which is less than 13.

   Substitute $b = 10$ into the original inequality:

   $3(10) - 7 < 32$

   $30 - 7 < 32$

   $23 < 32$

   This statement is true, so $b = 10$ is a valid solution.

2. Test a value outside the solution range: Let's choose $b = 15$, which is greater than 13.

   Substitute $b = 15$ into the original inequality:

   $3(15) - 7 < 32$

   $45 - 7 < 32$

   $38 < 32$

   This statement is false, so $b = 15$ is not a solution.

The verification process confirms that our solution, $b < 13$, is correct. By testing values within and outside the solution range, we gain confidence in the accuracy of our algebraic manipulations and the resulting solution. This step is an essential part of the problem-solving process, ensuring that we have arrived at the correct answer and understood the implications of the solution.

When solving inequalities, it's essential to be mindful of potential errors. One common mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This mistake can lead to an incorrect solution set. Another common error is misinterpreting the inequality symbols. For example, confusing less than (<) with less than or equal to (≤) can lead to including or excluding the boundary value in the solution set. It's also important to perform algebraic operations correctly, ensuring that the same operation is applied to both sides of the inequality to maintain balance. Careful attention to detail and a thorough understanding of the rules governing inequalities are crucial for avoiding these mistakes and arriving at the correct solution.

Inequalities are not just abstract mathematical concepts; they have numerous real-world applications. They are used in various fields, including economics, engineering, and computer science, to model and solve problems involving constraints and limitations. For instance, in economics, inequalities can be used to represent budget constraints, showing the range of goods and services a consumer can afford given their income. In engineering, inequalities can be used to design structures that can withstand certain loads or to optimize resource allocation within a system. In computer science, inequalities are used in algorithm design and analysis, particularly in areas like optimization and resource management. Understanding inequalities provides a valuable tool for analyzing and solving real-world problems that involve constraints and limitations. The ability to translate real-world scenarios into mathematical inequalities and solve them is a crucial skill in many disciplines.

Solving the inequality $3b - 7 < 32$ demonstrates the fundamental principles of solving linear inequalities. By following a step-by-step approach, we can isolate the variable and determine the solution set. Remember to maintain the balance of the inequality by performing the same operations on both sides and to reverse the inequality sign when multiplying or dividing by a negative number. The solution, $b < 13$, represents all values less than 13 and can be visualized on a number line. Inequalities are powerful tools with wide-ranging applications, making their understanding essential for various fields. By mastering the techniques for solving inequalities, we equip ourselves with valuable problem-solving skills that can be applied to real-world scenarios.