Solution To (-3/7)m < 21 A Step-by-Step Guide
In the realm of mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Inequalities, unlike equations, involve comparing expressions using symbols such as "less than" (<), "greater than" (>), "less than or equal to" (≤), and "greater than or equal to" (≥). Mastering the techniques for solving inequalities is essential for various mathematical applications, including optimization problems, calculus, and real-world scenarios involving constraints and limitations.
This comprehensive guide delves into the process of solving the inequality (-3/7)m < 21, providing a step-by-step approach that not only arrives at the solution but also elucidates the underlying principles and concepts. We will explore the properties of inequalities, the rules for manipulating them, and the importance of considering the sign of the coefficient when multiplying or dividing. By the end of this guide, you will have a solid understanding of how to solve this specific inequality and be equipped to tackle a wide range of similar problems.
Understanding the Inequality (-3/7)m < 21
The inequality (-3/7)m < 21 presents a relationship between the variable 'm' and the constant 21. The expression (-3/7)m represents a fraction multiplied by the variable 'm', and the inequality symbol "<" indicates that the value of this expression must be strictly less than 21. Our goal is to isolate the variable 'm' and determine the range of values that satisfy this condition.
To achieve this, we need to perform algebraic manipulations while adhering to the rules that govern inequalities. One key principle to remember is that multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. This is a crucial concept that will come into play when solving our inequality.
Step-by-Step Solution
Let's embark on the journey of solving the inequality (-3/7)m < 21, breaking down each step with clear explanations and justifications.
Step 1: Isolate the Variable Term
Our initial goal is to isolate the term containing the variable 'm'. In this case, the term is (-3/7)m. To isolate it, we need to eliminate the fraction (-3/7) that is multiplying 'm'.
To do this, we can multiply both sides of the inequality by the reciprocal of (-3/7), which is (-7/3). However, we must remember the crucial rule: multiplying or dividing an inequality by a negative number reverses the inequality sign.
Therefore, we multiply both sides by (-7/3) and flip the "<" sign to ">":
(-7/3) * (-3/7)m > 21 * (-7/3)
Step 2: Simplify Both Sides
Now, let's simplify both sides of the inequality.
On the left side, (-7/3) * (-3/7) simplifies to 1, effectively isolating 'm':
m > 21 * (-7/3)
On the right side, we multiply 21 by (-7/3). To do this, we can first divide 21 by 3, which gives us 7. Then, we multiply 7 by -7, resulting in -49:
m > -49
Step 3: The Solution
We have successfully isolated 'm' and arrived at the solution:
m > -49
This solution signifies that any value of 'm' greater than -49 will satisfy the original inequality (-3/7)m < 21.
Understanding the Solution Set
The solution m > -49 represents a set of infinite numbers. It includes all numbers greater than -49, but not -49 itself. We can visualize this solution set on a number line. Imagine a number line extending from negative infinity to positive infinity. Locate -49 on the number line. The solution m > -49 is represented by all the numbers to the right of -49, extending towards positive infinity. We use an open circle at -49 to indicate that -49 is not included in the solution set.
Common Mistakes to Avoid
Solving inequalities requires careful attention to detail, and certain common mistakes can lead to incorrect solutions. Let's highlight a few pitfalls to avoid:
Forgetting to Reverse the Inequality Sign
The most critical mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This oversight can lead to a completely incorrect solution set. Always remember to flip the sign when dealing with negative multipliers or divisors.
Incorrectly Applying the Distributive Property
If the inequality involves expressions with parentheses, ensure you correctly apply the distributive property before proceeding with other operations. Mistakes in distribution can alter the entire equation and lead to an erroneous solution.
Arithmetic Errors
Even minor arithmetic errors can have a significant impact on the solution. Double-check your calculations, especially when dealing with fractions, negative numbers, and multiple operations. A small mistake can cascade through the steps and result in an incorrect answer.
Practical Applications of Inequalities
Inequalities are not just abstract mathematical concepts; they have numerous practical applications in various fields. Let's explore a few examples:
Optimization Problems
Inequalities are fundamental in optimization problems, where the goal is to find the maximum or minimum value of a function subject to certain constraints. These constraints are often expressed as inequalities.
Real-World Constraints
In real-world scenarios, inequalities are used to represent limitations and restrictions. For instance, a budget constraint might be expressed as an inequality, limiting the total amount of spending.
Interval Notation
Interval notation is a concise way to represent the solution set of an inequality. For example, the solution m > -49 can be expressed in interval notation as (-49, ∞). This notation indicates that the solution includes all numbers from -49 (exclusive) to positive infinity.
Conclusion
Solving the inequality (-3/7)m < 21 involves applying algebraic manipulations while adhering to the rules governing inequalities. The crucial step is remembering to reverse the inequality sign when multiplying or dividing by a negative number. By following a step-by-step approach, simplifying expressions, and carefully considering the direction of the inequality, we arrive at the solution m > -49. This solution represents a set of infinite numbers greater than -49, which can be visualized on a number line or expressed in interval notation.
Mastering the techniques for solving inequalities is essential for various mathematical applications and real-world problem-solving scenarios. By understanding the underlying principles, avoiding common mistakes, and practicing consistently, you can confidently tackle a wide range of inequality problems.
Therefore, the correct answer is C. m > -49. This detailed explanation not only provides the solution but also delves into the reasoning behind it, making it a valuable learning resource.
