Solving M/4 < 6.5 Inequality Step By Step Guide

In this comprehensive guide, we will walk through the process of solving the inequality $\frac{M}{4} < 6.5$. This type of problem is a fundamental concept in algebra, and mastering it is crucial for more advanced mathematical topics. Our goal is not only to find the solution but also to thoroughly check our answer and express the solution set in the form of an inequality. Let's dive in and break down each step.

To solve the inequality $\frac{M}{4} < 6.5$, our primary objective is to isolate the variable M. This means we want to get M by itself on one side of the inequality. The inequality states that M divided by 4 is less than 6.5. To undo the division, we need to perform the inverse operation, which is multiplication. We will multiply both sides of the inequality by 4. This ensures that we maintain the balance of the inequality, keeping it mathematically sound. When we multiply both sides by 4, we get:

$4 \cdot \frac{M}{4} < 6.5 \cdot 4$

The left side simplifies nicely because multiplying $\frac{M}{4}$ by 4 cancels out the division, leaving us with just M. On the right side, we multiply 6.5 by 4. To do this, we can think of 6.5 as 6 and a half. So, 6.5 multiplied by 4 is the same as (6 multiplied by 4) plus (0.5 multiplied by 4). Six times four is 24, and 0.5 (which is one-half) times 4 is 2. Adding these together, we get 24 + 2 = 26. Thus, our inequality now looks like:

$M < 26$

This inequality tells us that M is less than 26. This is our preliminary solution, but to ensure its accuracy, we need to check it. Checking our solution involves substituting values back into the original inequality to confirm that our result holds true. This step is vital because it helps us catch any potential errors made during the solving process. Mathematical operations, especially those involving inequalities, can sometimes lead to mistakes, and checking is our safeguard against such errors. Moreover, understanding how to check solutions reinforces the fundamental principles of inequality and algebraic manipulation. By verifying our solution, we gain confidence in our answer and a deeper understanding of the mathematical concepts involved.

To check our solution, we need to substitute a value for M that is less than 26 into the original inequality $\frac{M}{4} < 6.5$. This process validates whether our calculated solution (M < 26) satisfies the initial condition. The act of substituting a value helps ensure that no algebraic errors were made during the solution process. It also provides a practical understanding of what the inequality means. Let’s choose a value for M that is clearly less than 26. A straightforward choice is M = 24. This number is easy to work with and should give us a clear indication of whether our solution is correct. We will now substitute M = 24 into the original inequality:

$\frac{24}{4} < 6.5$

Now, we perform the division on the left side of the inequality. Twenty-four divided by 4 is 6. So, our inequality becomes:

$6 < 6.5$

This statement is true. Six is indeed less than 6.5. This confirms that our solution M < 26 is correct for at least this chosen value. However, to be even more confident in our solution, it’s beneficial to test another value. This helps ensure that our solution holds true across a range of values, not just one specific number. For our second check, let’s choose a value that is significantly less than 26, such as M = 0. Zero is often a good choice because it simplifies calculations, and it represents a clear case within our solution set. Substituting M = 0 into the original inequality, we get:

$\frac{0}{4} < 6.5$

Zero divided by 4 is 0, so the inequality simplifies to:

$0 < 6.5$

This statement is also true. Zero is less than 6.5. These two checks give us a high level of confidence that our solution M < 26 is correct. By using two different values, we have tested our solution under different conditions, reinforcing the validity of our answer. The process of checking solutions is a critical step in solving inequalities and equations. It not only verifies the correctness of the answer but also deepens our understanding of the mathematical principles involved. This practice is highly recommended for all mathematical problem-solving, as it helps prevent errors and builds a solid foundation in mathematical thinking.

Having solved the inequality and thoroughly checked our solution, the next crucial step is to write the solution set as an inequality. This final step ensures that we clearly and accurately communicate the range of values that satisfy the original condition. The solution set represents all possible values of M that make the inequality $\frac{M}{4} < 6.5$ true. We have already determined that M must be less than 26. This means that any number smaller than 26 will satisfy the inequality. To express this mathematically, we use the “less than” symbol, which gives us the inequality:

$M < 26$

This inequality succinctly states that the solution set includes all real numbers less than 26. It is a clear and concise way of representing the range of possible values for M. The solution set does not include 26 itself, because the inequality is strictly “less than” and not “less than or equal to.” If the inequality were $M \leq 26$, then 26 would be included in the solution set. However, in our case, 26 is the upper bound, but it is not part of the solution itself.

Understanding how to express the solution set as an inequality is fundamental in algebra. It provides a precise way to describe an infinite number of solutions. In many real-world applications, inequalities are used to model constraints and conditions. For example, a budget constraint might be expressed as an inequality, where the total spending must be less than or equal to a certain amount. Similarly, in physics, inequalities can be used to describe the range of possible values for physical quantities, such as speed or temperature. The ability to solve and express solution sets for inequalities is therefore a valuable skill in various fields.

In summary, we have successfully solved the inequality $\frac{M}{4} < 6.5$, checked our solution using multiple values, and expressed the solution set as the inequality $M < 26$. This comprehensive approach not only provides the correct answer but also ensures a deep understanding of the process and the underlying mathematical concepts. Solving inequalities is a key skill in mathematics, and mastering this skill will open doors to more advanced topics and real-world applications. The careful, step-by-step method we have used here can be applied to a wide range of inequality problems, helping you to solve them accurately and confidently.

In conclusion, solving the inequality $\frac{M}{4} < 6.5$ involves isolating the variable, checking the solution, and expressing the solution set as an inequality. By following a systematic approach, we can accurately determine that the solution set is $M < 26$. This process reinforces essential algebraic skills and provides a solid foundation for tackling more complex problems. Remember to always check your solutions to ensure accuracy and deepen your understanding of the underlying concepts.