Solving Inequalities Finding The Range Of 'm' And Representing On A Number Line

Introduction

This article delves into the fascinating world of inequalities, focusing on determining the range of values for the variable 'm' that satisfy a given inequality. We will explore the step-by-step process of solving the inequality $\frac{m-1}{3}+\frac{m+3}{7} \geq m-2$, and then visually represent the solution set on a number line. Understanding inequalities is crucial in various fields of mathematics and real-world applications, from optimization problems to financial modeling. So, let's embark on this journey of mathematical exploration and unravel the mysteries of inequalities.

The Importance of Understanding Inequalities

Inequalities, unlike equations, deal with relationships where values are not necessarily equal. They are fundamental in expressing constraints, limitations, or ranges of possibilities. In mathematics, inequalities form the basis for concepts like intervals, optimization, and calculus. In real-world scenarios, inequalities are used to model budget constraints, resource allocation, and decision-making under uncertainty. For example, a business might use inequalities to determine the minimum production level needed to achieve a certain profit margin, or an engineer might use them to ensure that a structure can withstand a certain range of loads. Therefore, mastering the art of solving inequalities is not just an academic exercise but a valuable skill with practical implications.

Solving the Inequality Step-by-Step

Step 1: Eliminating Fractions

To solve the inequality $\frac{m-1}{3}+\frac{m+3}{7} \geq m-2$, the first step is to eliminate the fractions. This makes the inequality easier to manipulate and solve. We achieve this by finding 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 clearing the fractions. The key is to multiply both sides by the same positive number; multiplying by a negative number would require flipping the inequality sign. Performing this multiplication, we get:

$$21 \left( \frac{m-1}{3}+\frac{m+3}{7} \right) \geq 21(m-2)$$

Step 2: Distributing and Simplifying

Next, we distribute the 21 on both sides of the inequality. On the left side, we distribute the 21 to both terms within the parentheses. This involves dividing 21 by the denominators of the fractions. On the right side, we simply distribute the 21 to both terms within the parentheses. After distributing, we simplify the resulting expression by performing the necessary multiplications and cancellations. This step is crucial for isolating the variable 'm' and bringing us closer to the solution. The simplified inequality will look like this:

$$7(m-1) + 3(m+3) \geq 21m - 42$$

Step 3: Expanding Parentheses

Now, we expand the parentheses on the left side of the inequality. This involves multiplying the constants outside the parentheses by each term inside the parentheses. This step is a straightforward application of the distributive property. Expanding the parentheses helps us to combine like terms and further simplify the inequality. Remember to pay close attention to the signs when multiplying. After expanding, we have:

$$7m - 7 + 3m + 9 \geq 21m - 42$$

Step 4: Combining Like Terms

In this step, we combine the like terms on both sides of the inequality. Like terms are terms that have the same variable raised to the same power. On the left side, we combine the 'm' terms (7m and 3m) and the constant terms (-7 and 9). On the right side, there are no like terms to combine. Combining like terms simplifies the inequality and makes it easier to isolate the variable 'm'. The result of this step is:

$$10m + 2 \geq 21m - 42$$

Step 5: Isolating the Variable

To isolate the variable 'm', we need to move all the 'm' terms to one side of the inequality and all the constant terms to the other side. We can do this by adding or subtracting terms from both sides of the inequality. In this case, we subtract 10m from both sides and add 42 to both sides. This will effectively move all the 'm' terms to the right side and all the constant terms to the left side. Remember that performing the same operation on both sides of the inequality maintains the inequality. After isolating the variable, we get:

$$44 \geq 11m$$

Step 6: Solving for 'm'

Finally, to solve for 'm', we divide both sides of the inequality by 11. Since we are dividing by a positive number, we do not need to flip the inequality sign. Dividing both sides by 11 isolates 'm' and gives us the solution to the inequality. The solution is:

$$m \leq 4$$

This means that the inequality holds true for all values of 'm' that are less than or equal to 4.

Representing the Solution on a Number Line

Understanding Number Line Representation

A number line is a visual tool used to represent real numbers. It is a horizontal line with numbers placed at equal intervals. Representing the solution of an inequality on a number line provides a clear visual understanding of the range of values that satisfy the inequality. For an inequality like $m \leq 4$, we represent all the numbers less than or equal to 4 on the number line. This involves drawing a line or shading the portion of the number line that corresponds to the solution set. The number line representation helps to visualize the solution set and understand the concept of intervals.

Representing $m \leq 4$ on the Number Line

To represent $m \leq 4$ on the number line, we first locate the number 4 on the number line. Since the inequality includes 'equal to' ($\leq$), we use a closed circle or a filled-in dot at 4 to indicate that 4 is included in the solution set. Then, we shade or draw a line to the left of 4, indicating that all numbers less than 4 are also part of the solution set. This shaded region represents the range of values of 'm' that satisfy the inequality. The number line visually demonstrates that any value of 'm' from negative infinity up to and including 4 is a solution to the inequality.

Visual Interpretation

The number line representation provides an intuitive understanding of the solution. The closed circle at 4 signifies that 4 is a boundary point and is included in the solution. The shaded region extending to the left indicates that the solution set is unbounded on the negative side, meaning there is no lower limit to the values of 'm' that satisfy the inequality. This visual representation is a powerful tool for understanding and communicating the solution of inequalities.

Conclusion

Recap of the Solution

In this article, we embarked on a journey to solve the inequality $\frac{m-1}{3}+\frac{m+3}{7} \geq m-2$. We meticulously followed a step-by-step process, starting with eliminating fractions, simplifying the inequality, isolating the variable 'm', and finally arriving at the solution: $m \leq 4$. This solution signifies that any value of 'm' that is less than or equal to 4 will satisfy the original inequality. We then visually represented this solution on a number line, providing a clear and intuitive understanding of the range of possible values for 'm'.

Importance of Mastering Inequalities

Understanding and solving inequalities is a fundamental skill in mathematics and has wide-ranging applications in various fields. Inequalities are used to model constraints, optimize solutions, and make informed decisions in real-world scenarios. Mastering inequalities enhances problem-solving abilities and analytical thinking. The ability to manipulate and solve inequalities is crucial for success in higher-level mathematics and in professions that require quantitative analysis. Therefore, a solid grasp of inequalities is an invaluable asset.

Final Thoughts

Solving inequalities is not just about finding a numerical answer; it's about understanding the relationships between quantities and the constraints that govern them. The process of solving an inequality involves careful manipulation, attention to detail, and a clear understanding of the underlying principles. The number line representation adds a visual dimension to the solution, making it easier to grasp the concept of a solution set. By mastering these techniques, you can confidently tackle a wide range of problems involving inequalities and apply them to real-world situations.