Finding The Range Of M And Representing Solutions On A Number Line

This article delves into the process of finding the range of values for m that satisfy the inequality (m-1)/3 + (m+3)/7 ≥ m-2. We will explore the step-by-step algebraic manipulations required to isolate m and determine the solution set. Furthermore, we will visually represent the solution on a number line, providing a clear understanding of the values of m that fulfill the given condition.

Unveiling the Solution: A Step-by-Step Approach

To begin, let's break down the process of solving the inequality. Our primary goal is to isolate m on one side of the inequality. This involves a series of algebraic operations, ensuring that we maintain the balance of the inequality at each step.

1. Eliminating Fractions: The initial step involves getting rid of the fractions to simplify the expression. We can achieve this by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators, which in this case is 21. This yields:

   7(m-1) + 3(m+3) ≥ 21(m-2)

2. Expanding the Expressions: Next, we expand the expressions on both sides of the inequality:

   7m - 7 + 3m + 9 ≥ 21m - 42

3. Combining Like Terms: Now, we combine the like terms on each side:

   10m + 2 ≥ 21m - 42

4. Isolating m: Our aim is to get all the m terms on one side and the constant terms on the other. Subtracting 10m from both sides gives:

   2 ≥ 11m - 42

   Adding 42 to both sides results in:

   44 ≥ 11m

5. Solving for m: Finally, we divide both sides by 11 to isolate m:

   4 ≥ m

   This can also be written as:

   m ≤ 4

Therefore, the solution to the inequality is m ≤ 4. This means that any value of m that is less than or equal to 4 will satisfy the original inequality.

Visualizing the Solution: Representing on a Number Line

A number line provides a visual representation of the solution set. To represent m ≤ 4 on a number line, we draw a horizontal line and mark the point 4. Since the inequality includes m being equal to 4, we use a closed circle (or a filled-in dot) at 4. Then, we shade the portion of the number line to the left of 4, indicating all values less than 4.

This visual representation clearly shows that the solution includes 4 and all numbers extending infinitely to the left on the number line. This graphical method provides an intuitive understanding of the solution set.

Key Concepts and Considerations

• Inequality Properties: When working with inequalities, it's crucial to remember the properties of inequalities. For instance, multiplying or dividing both sides by a negative number reverses the direction of the inequality sign.
• Solution Set: The solution to an inequality is a set of values, rather than a single value as in an equation. This set can be represented in various ways, including inequality notation (e.g., m ≤ 4), interval notation (e.g., (-∞, 4]), and graphically on a number line.
• Number Line Representation: The number line provides a visual tool for understanding the solution set. It helps to identify the range of values that satisfy the inequality.

In this section, we will provide a more in-depth explanation of each step involved in solving the inequality, highlighting the underlying mathematical principles and potential pitfalls.

Step 1: Eliminating Fractions – A Detailed Look

The initial step of multiplying both sides of the inequality by the least common multiple (LCM) is crucial for simplifying the expression. The LCM of the denominators (3 and 7) is 21. This ensures that when we multiply each fraction by 21, the denominators will cancel out, leaving us with integer coefficients. This eliminates the complexities associated with fractions and makes the subsequent algebraic manipulations easier.

Mathematically, this step is justified by the multiplicative property of inequality, which states that multiplying both sides of an inequality by a positive number preserves the inequality. Since 21 is positive, we maintain the ≥ sign.

For example:

21 * [(m-1)/3 + (m+3)/7] ≥ 21 * (m-2)

This results in:

7(m-1) + 3(m+3) ≥ 21(m-2)

Potential Pitfalls: It's essential to ensure that the LCM is correctly identified and that the multiplication is distributed accurately to each term on both sides of the inequality. A common mistake is forgetting to multiply the constant term (-2) on the right side by 21.

Step 2: Expanding the Expressions – Applying the Distributive Property

Expanding the expressions involves applying the distributive property, which states that a(b + c) = ab + ac. This step is necessary to remove the parentheses and combine like terms. We carefully multiply each term inside the parentheses by the coefficient outside.

For example:

7(m-1) becomes 7m - 71 = 7m - 7

Similarly,

3(m+3) becomes 3m + 33 = 3m + 9

And,

21(m-2) becomes 21m - 212 = 21m - 42

Potential Pitfalls: Care should be taken to correctly apply the distributive property, especially when dealing with negative signs. For instance, a negative sign outside the parentheses changes the signs of all terms inside the parentheses.

Step 3: Combining Like Terms – Simplifying the Inequality

Combining like terms involves grouping terms with the same variable and constant terms separately. This simplifies the inequality and makes it easier to isolate m. We add or subtract coefficients of the same variable and combine constant terms.

For example, on the left side of the inequality:

7m + 3m becomes 10m

And,

-7 + 9 becomes 2

Thus, the left side simplifies to 10m + 2.

On the right side, there are no like terms to combine, so it remains 21m - 42.

Potential Pitfalls: A common error is incorrectly combining terms, such as adding or subtracting terms with different variables. It's crucial to only combine terms that have the same variable raised to the same power.

Step 4: Isolating m – Rearranging Terms

Isolating m involves rearranging the terms so that all terms containing m are on one side of the inequality and constant terms are on the other side. This is achieved by adding or subtracting the same term from both sides of the inequality, maintaining the balance.

In our example, we subtract 10m from both sides to get all m terms on the right side:

10m + 2 - 10m ≥ 21m - 42 - 10m

This simplifies to:

2 ≥ 11m - 42

Next, we add 42 to both sides to isolate the m term:

2 + 42 ≥ 11m - 42 + 42

This simplifies to:

44 ≥ 11m

Potential Pitfalls: When adding or subtracting terms, it's crucial to perform the operation on both sides of the inequality to maintain its balance. Also, pay attention to the signs of the terms being added or subtracted.

Step 5: Solving for m – The Final Step

Finally, we solve for m by dividing both sides of the inequality by the coefficient of m. In our case, the coefficient of m is 11. We divide both sides by 11:

44 / 11 ≥ 11m / 11

This simplifies to:

4 ≥ m

Which can also be written as:

m ≤ 4

This means that m is less than or equal to 4. This is the solution set to the inequality.

Potential Pitfalls: If we were to divide both sides by a negative number, we would need to reverse the direction of the inequality sign. This is a crucial rule to remember when working with inequalities.

A number line is a powerful tool for visualizing the solution set of an inequality. It provides a graphical representation of all the values that satisfy the inequality. To represent m ≤ 4 on a number line, we follow these steps:

1. Draw a Horizontal Line: Draw a straight horizontal line to represent the number line.
2. Mark the Critical Point: Identify the critical point, which is the value of m where the inequality changes direction. In our case, the critical point is 4. Mark 4 on the number line.
3. Use a Closed or Open Circle: If the inequality includes equality (≤ or ≥), use a closed circle (filled-in dot) at the critical point to indicate that the point is included in the solution set. If the inequality does not include equality (< or >), use an open circle to indicate that the point is not included.
4. Shade the Solution Region: Shade the portion of the number line that represents the solution set. For m ≤ 4, we shade the region to the left of 4, including 4 itself.

The number line representation clearly shows that the solution includes 4 and all numbers extending infinitely to the left. This visual aid provides an intuitive understanding of the solution set.

In this article, we have thoroughly explored the process of solving the inequality (m-1)/3 + (m+3)/7 ≥ m-2. We have broken down the solution into step-by-step algebraic manipulations, highlighting the key concepts and potential pitfalls at each stage. Furthermore, we have demonstrated how to represent the solution on a number line, providing a visual understanding of the values of m that satisfy the inequality.

By understanding the underlying mathematical principles and practicing these techniques, you can confidently tackle similar inequality problems and gain a deeper appreciation for the power of algebraic reasoning. The ability to solve inequalities is a fundamental skill in mathematics and has wide-ranging applications in various fields, from science and engineering to economics and finance. Mastering these concepts will undoubtedly strengthen your problem-solving abilities and pave the way for more advanced mathematical studies.

Now, let's address the second part of the question: representing the solution on a number line. This visual representation is a powerful tool for understanding the range of values that m can take while satisfying the inequality. The solution we found, m ≤ 4, signifies that any value of m less than or equal to 4 will make the inequality true. To depict this on a number line, we follow a few key steps:

1. Draw the Number Line: Start by drawing a straight horizontal line. This line represents the entire spectrum of real numbers, extending infinitely in both positive and negative directions. It's crucial to have a clear and well-drawn line to accurately represent the solution set.

2. Mark the Critical Point: The critical point in our solution is 4. This is the value where the inequality transitions from being true to false. Locate 4 on the number line and mark it prominently. The position of this point is crucial as it acts as a boundary for the solution set. The accuracy in placing this point directly impacts the clarity of the visual representation.

3. Use a Closed Circle: Because our solution includes m being equal to 4 (m ≤ 4), we use a closed circle (a filled-in dot) at the point 4. This closed circle indicates that 4 itself is part of the solution set. If the inequality were m < 4, we would use an open circle to show that 4 is not included in the solution. This subtle difference in notation is vital for correctly interpreting the solution.

4. Shade the Solution Region: Since m can be any value less than or equal to 4, we shade the portion of the number line to the left of 4. This shaded region visually represents all the numbers that satisfy the inequality. The shading extends from 4 towards negative infinity, illustrating the infinite range of possible values for m. This step is perhaps the most intuitive part of the representation, as it gives a clear visual of the solution's extent.

Interpretation of the Number Line Representation

The number line now provides a clear visual summary of the solution. The closed circle at 4 tells us that 4 is included in the solution, and the shaded region extending to the left indicates that all numbers less than 4 are also solutions. For instance, values like 3, 0, -2, and -100 are all part of the solution set. This representation is invaluable because it instantly conveys the range of permissible values for m in a way that a simple inequality statement might not.

Why Number Line Representation Matters

Visualizing solutions on a number line is not just a matter of graphical representation; it's a fundamental skill in understanding inequalities. It helps in:

• Conceptual Understanding: The number line provides a concrete way to grasp the concept of a range of solutions, rather than a single value. This is particularly useful when dealing with inequalities, where the solution is often a set of numbers.
• Problem Solving: Visualizing the solution can aid in solving more complex problems involving multiple inequalities or compound inequalities. By representing each inequality on a number line, it becomes easier to identify the overlapping solution sets.
• Communication: The number line is an effective way to communicate the solution to others, as it offers a clear and unambiguous representation. It's a universal tool in mathematics for expressing solution sets.

Common Mistakes to Avoid

While representing solutions on a number line is straightforward, some common mistakes can lead to misinterpretations:

• Using the Wrong Circle Type: Forgetting to use a closed circle for inequalities that include equality (≤ or ≥) or using a closed circle when an open circle is required (< or >) is a frequent error. Always double-check the inequality sign.
• Shading the Wrong Region: It's essential to shade the correct portion of the number line. If the solution is m ≤ 4, shade to the left; if it's m ≥ 4, shade to the right. A simple way to remember this is to think about which direction corresponds to smaller or larger values.
• Inaccurate Point Placement: Marking the critical point at the wrong location can distort the entire representation. Ensure the point is placed accurately on the number line.

Practice Makes Perfect

The best way to master number line representation is through practice. Work through various examples with different inequalities to become comfortable with the notation and the process of visualizing solutions. As you gain experience, you'll find that the number line becomes an indispensable tool in your mathematical toolkit. The more you practice, the better you'll become at quickly and accurately translating inequalities into visual representations, enhancing both your understanding and problem-solving abilities.

In conclusion, representing the solution m ≤ 4 on a number line involves drawing the line, marking the critical point 4 with a closed circle, and shading the region to the left. This visual representation provides a clear and intuitive understanding of the solution set, highlighting the range of values that satisfy the given inequality. This skill is not only crucial for solving mathematical problems but also for fostering a deeper understanding of mathematical concepts and their applications. Through practice and attention to detail, number line representation becomes a powerful tool in mathematical analysis and communication.