Solving Inequalities Finding The Range Of M For (m-1)/3 + (m+1)/7 ≥ M-2

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In the realm of mathematics, inequalities serve as powerful tools for expressing relationships where quantities are not necessarily equal. They allow us to define ranges of values that satisfy specific conditions, opening doors to problem-solving in various fields, from optimization to economics. This article delves into the intricacies of solving inequalities, offering a step-by-step guide illustrated with a detailed example. We will explore the fundamental principles, techniques, and nuances involved in finding the solution sets that make inequalities true. By the end of this exploration, you will be equipped with the knowledge and skills to tackle a wide array of inequality problems, empowering you to make informed decisions and gain deeper insights into mathematical relationships.

Understanding Inequalities

Inequalities, unlike equations, express relationships where two quantities are not necessarily equal. They use symbols such as 'greater than' (>), 'less than' (<), 'greater than or equal to' (≥), and 'less than or equal to' (≤) to compare values. Solving inequalities involves finding the range of values that satisfy the given condition. This contrasts with equations, where we typically seek specific numerical solutions. Inequalities often arise in real-world scenarios where constraints or limitations exist, making them essential tools in optimization problems, resource allocation, and decision-making processes. For instance, an inequality might represent the maximum weight a bridge can support or the minimum profit a business needs to achieve. Understanding how to manipulate and solve inequalities is crucial for modeling and analyzing these situations effectively.

Linear Inequalities: The Building Blocks

At the heart of inequality solving lies the concept of linear inequalities. These are inequalities that involve linear expressions, meaning the variables are raised to the power of 1. They form the foundation for understanding more complex inequalities and serve as the basis for many mathematical models. A typical linear inequality might look like 2x + 3 < 7, where 'x' is the variable we want to find the range of values for. Solving linear inequalities involves similar steps to solving linear equations, with one crucial difference: multiplying or dividing by a negative number reverses the inequality sign. This rule is paramount and stems from the properties of the number line. For example, if we have -2 < 4, multiplying by -1 gives 2 > -4, which maintains the truth of the statement. Mastering the techniques for solving linear inequalities is crucial because they often appear as components of larger problems, such as systems of inequalities or optimization problems.

The Golden Rule: Preserve the Truth

The key to solving inequalities lies in the principle of preserving the truth. Any operation performed on an inequality must maintain the validity of the relationship. This means that adding or subtracting the same quantity from both sides, or multiplying or dividing by a positive quantity, does not change the inequality's direction. However, multiplying or dividing by a negative quantity necessitates flipping the inequality sign. This rule is a cornerstone of inequality manipulation and ensures that the solution set remains accurate. To illustrate, consider the inequality x + 3 > 5. Subtracting 3 from both sides gives x > 2, which preserves the solution set. But if we have -2x < 4, dividing by -2 gives x > -2, flipping the sign because we divided by a negative number. This careful attention to the rules ensures that the solution obtained is consistent with the original inequality, preventing errors and leading to correct interpretations.

a) Determining the Range of m: A Step-by-Step Solution

Now, let's tackle the inequality $\frac{m-1}{3}+\frac{m+1}{7} \geq m-2$ head-on. Our mission is to find the range of values for m that makes this inequality true. This involves a series of algebraic manipulations, each carefully designed to isolate 'm' and reveal the solution set. We'll start by clearing the fractions, then simplify the expression, and finally, isolate 'm' to find its range. This process not only provides the answer but also showcases the power of algebraic techniques in solving mathematical problems. The solution set will give us a clear picture of the values of 'm' that satisfy the given condition, which is essential for various applications, from optimization problems to real-world scenarios where constraints are represented by inequalities.

Step 1: Clearing the Fractions

Fractions can often complicate algebraic expressions, so our first step is to eliminate them. To do this, we find the least common multiple (LCM) of the denominators, which in this case are 3 and 7. The LCM of 3 and 7 is 21. We then multiply both sides of the inequality by 21. This ensures that we maintain the inequality while removing the fractions, making the equation easier to work with. Multiplying both sides by the LCM is a standard technique in solving equations and inequalities involving fractions, as it simplifies the expression and allows us to proceed with other algebraic manipulations. This step is crucial for reducing the complexity of the inequality and paving the way for isolating the variable.

Multiplying both sides of the inequality by 21, we get:

21∗(m−13+m+17)≥21∗(m−2)21 * (\frac{m-1}{3}+\frac{m+1}{7}) \geq 21 * (m-2)

This simplifies to:

7(m−1)+3(m+1)≥21(m−2)7(m-1) + 3(m+1) \geq 21(m-2)

Step 2: Simplifying the Expression

Now that we've cleared the fractions, it's time to simplify the inequality by expanding the brackets and combining like terms. This step involves distributing the constants and then collecting terms with 'm' and constant terms. Simplifying the expression makes it easier to identify the key components and isolate the variable. It's a crucial step in reducing the complexity of the inequality and revealing the underlying relationship between the variable and the constants. Accurate simplification is essential for avoiding errors and obtaining the correct solution set. This step often involves basic algebraic operations but requires careful attention to detail to ensure the expression is correctly simplified.

Expanding the brackets, we have:

7m−7+3m+3≥21m−427m - 7 + 3m + 3 \geq 21m - 42

Combining like terms, we get:

10m−4≥21m−4210m - 4 \geq 21m - 42

Step 3: Isolating m

The ultimate goal is to isolate 'm' on one side of the inequality. This involves a series of algebraic manipulations to group the 'm' terms and the constant terms separately. We'll move the 'm' terms to one side and the constants to the other, ensuring we maintain the inequality's direction. Isolating the variable is the key step in solving any inequality, as it reveals the range of values that satisfy the condition. This process requires careful application of algebraic principles and attention to detail to avoid errors. Once 'm' is isolated, we can clearly see the solution set and understand the values that make the inequality true.

Subtracting 10m from both sides:

−4≥11m−42-4 \geq 11m - 42

Adding 42 to both sides:

38≥11m38 \geq 11m

Dividing both sides by 11:

3811≥m\frac{38}{11} \geq m

Thus, $m \leq \frac{38}{11}$.

b) Representing the Solution on a Number Line

A number line provides a visual representation of the solution set for an inequality. It's a powerful tool for understanding the range of values that satisfy the condition and for communicating the solution clearly. To represent the solution on a number line, we draw a line and mark the critical value, which in this case is $ rac{38}{11}$. Since the inequality is 'less than or equal to', we use a closed circle at this point to indicate that it's included in the solution set. Then, we shade the region to the left of this point, representing all the values less than $\frac{38}{11}$. This visual representation makes it easy to see the solution set and understand the range of values that satisfy the inequality. It's a valuable tool for both solving and interpreting inequalities.

Visualizing the Solution

To represent the solution $m \leq \frac{38}{11}$ on a number line, we draw a horizontal line and mark the point $ rac{38}{11}$. Since the inequality includes 'equal to', we use a closed circle at this point. Then, we shade the region to the left of $\frac{38}{11}$, indicating all values less than or equal to $\frac{38}{11}$. This visual representation provides a clear understanding of the solution set and helps to interpret the inequality's meaning. The number line is an invaluable tool for visualizing solutions and gaining insights into mathematical relationships.

Conclusion: Mastering Inequalities

In conclusion, solving inequalities is a fundamental skill in mathematics with far-reaching applications. By understanding the principles of inequality manipulation, we can tackle a wide range of problems, from simple linear inequalities to more complex scenarios. The key lies in preserving the truth of the relationship and using algebraic techniques to isolate the variable. Representing the solution on a number line provides a visual aid for understanding the range of values that satisfy the inequality. With practice and a solid grasp of the concepts, you can master inequalities and unlock their power in problem-solving.

By working through the example of $\frac{m-1}{3}+\frac{m+1}{7} \geq m-2$, we have demonstrated a step-by-step approach to solving inequalities. This method involves clearing fractions, simplifying expressions, isolating the variable, and representing the solution on a number line. These techniques are applicable to a variety of inequality problems and provide a solid foundation for further mathematical exploration. Mastering these skills will empower you to approach inequality problems with confidence and clarity.

This journey into solving inequalities has equipped you with the tools and knowledge to tackle a wide range of mathematical challenges. Remember, the key to success lies in understanding the principles, practicing the techniques, and visualizing the solutions. With dedication and perseverance, you can master inequalities and unlock their power in problem-solving and decision-making.