Solving $-5 -5 \leq \frac{4-3m}{2} < 1$ A Step-by-Step Guide

Understanding Compound Inequalities

Compound inequalities, such as the one presented here, $-5 o -5 \leq \frac{4-3m}{2} < 1$, are mathematical statements that combine two or more inequalities. Effectively solving these inequalities requires a systematic approach, ensuring each step maintains the integrity of the inequality while isolating the variable. In this article, we will dissect the given compound inequality step-by-step, providing a clear and comprehensive solution. The most crucial aspect of dealing with inequalities is the understanding that operations performed on one part of the inequality must be mirrored on all other parts to preserve the relationship. Furthermore, multiplying or dividing by a negative number necessitates flipping the inequality sign, a detail often overlooked but vital for accurate results. The solution set for a compound inequality often represents a range of values, and this range can be visually represented on a number line. This visual representation aids in comprehending the solution and identifying any potential errors in the algebraic manipulation. Therefore, a solid grasp of inequality properties and careful application of algebraic principles are paramount for solving compound inequalities successfully.

Step-by-Step Solution

Let's break down the solution to the compound inequality $-5 \leq \frac{4-3m}{2} < 1$. Our primary goal is to isolate the variable m in the middle section of the inequality. This involves a series of algebraic manipulations applied consistently across all parts of the inequality. We'll start by eliminating the fraction, then deal with the constants, and finally isolate m itself. Each step will be explained in detail to ensure clarity and understanding. Remember, the key is to maintain balance and apply operations uniformly to all segments of the inequality. Throughout this process, we'll also highlight the critical rule of flipping the inequality sign when multiplying or dividing by a negative number. This step is often a source of error, so we'll emphasize its importance and demonstrate its application within the context of our problem. By meticulously following each step, we can arrive at the correct solution set for m. This methodical approach is not only crucial for this particular problem but also serves as a template for solving various compound inequalities.

1. Eliminate the Fraction

To eliminate the fraction, we multiply all parts of the inequality by 2:

$2 * -5 \leq 2 * \frac{4-3m}{2} < 2 * 1$

This simplifies to:

$-10 \leq 4 - 3m < 2$

2. Isolate the Term with m

Next, we subtract 4 from all parts of the inequality to isolate the term with m:

$-10 - 4 \leq 4 - 3m - 4 < 2 - 4$

This gives us:

$-14 \leq -3m < -2$

3. Isolate m

Now, we divide all parts of the inequality by -3. Crucially, because we are dividing by a negative number, we must reverse the direction of the inequality signs:

$\frac{-14}{-3} -3m/-3 > \frac{-2}{-3}$

This simplifies to:

$\frac{14}{3} m > \frac{2}{3}$

4. Rewrite the Inequality

It's conventional to write inequalities with the variable on the left. So, we rewrite the inequality, remembering to flip the direction again to maintain correctness:

$\frac{2}{3} < m \leq \frac{14}{3}$

Solution and Interpretation

The solution to the compound inequality $-5 \leq \frac{4-3m}{2} < 1$ is $\frac{2}{3} < m \leq \frac{14}{3}$. This means that m can be any value greater than $\frac{2}{3}$ and less than or equal to $\frac{14}{3}$.

Number Line Representation

To visualize this solution, we can represent it on a number line. We would place an open circle at $\frac{2}{3}$ (since m is strictly greater than this value) and a closed circle at $\frac{14}{3}$ (since m can be equal to this value). The region between these two points, including the closed circle at $\frac{14}{3}$, represents the solution set. This graphical representation provides an intuitive understanding of the range of values that satisfy the original inequality.

Alternative Representations

The solution can also be expressed in interval notation as $(\frac{2}{3}, \frac{14}{3}]$. This notation clearly indicates the lower bound, upper bound, and whether each bound is included in the solution set (parentheses for exclusive, brackets for inclusive). Understanding these different representations – inequality notation, number line representation, and interval notation – is crucial for a comprehensive grasp of inequality solutions.

Common Mistakes and How to Avoid Them

Solving compound inequalities can be tricky, and there are common pitfalls that students often encounter. Recognizing these mistakes and understanding how to avoid them is essential for accuracy and confidence in solving these types of problems. The most frequent error is forgetting to flip the inequality sign when multiplying or dividing by a negative number. This can lead to a completely incorrect solution set. Another common mistake is not applying the same operation to all parts of the inequality, disrupting the balance and invalidating the result. Furthermore, careless arithmetic errors during the simplification process can also lead to wrong answers.

1. Forgetting to Flip the Inequality Sign

The most critical mistake, as mentioned earlier, is forgetting to reverse the inequality signs when multiplying or dividing by a negative number. Let's illustrate this with our example. When we divided by -3, we had: $-14 \leq -3m < -2$. Dividing directly without flipping would incorrectly yield $\frac{14}{3} \leq m < \frac{2}{3}$, a completely different and wrong solution. Always double-check this step when dealing with negative coefficients.

2. Unequal Operations

Ensure that any operation performed is applied to every part of the inequality. For instance, if you subtract 4, subtract it from all three segments. Failing to do so distorts the relationships and leads to an incorrect solution. Maintaining balance is the golden rule of inequality manipulation.

3. Arithmetic Errors

Careless arithmetic mistakes are surprisingly common. A simple sign error or miscalculation can throw off the entire solution. It's always a good practice to double-check your arithmetic, especially when dealing with fractions or negative numbers. Writing out each step clearly and methodically can also help minimize these errors.

Best Practices to Avoid Mistakes

To minimize these errors, adopt a systematic approach: (1) Write each step clearly and legibly. (2) Double-check your arithmetic calculations. (3) Pay extra attention when multiplying or dividing by negative numbers and remember to flip the inequality signs. (4) After obtaining a solution, consider testing a value within your solution range in the original inequality to verify its validity. This is an excellent way to catch errors and build confidence in your answer.

Real-World Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have numerous applications in real-world scenarios. Understanding and solving inequalities is crucial in various fields, including economics, engineering, and computer science. They are used to model constraints, optimize solutions, and make informed decisions. For example, inequalities can be used to determine the range of production levels that maximize profit, the safety limits of a structure under stress, or the resource allocation in a network system. By exploring these real-world applications, we can gain a deeper appreciation for the practical significance of inequalities.

1. Economics and Business

In economics, inequalities are frequently used to model budget constraints and profit maximization problems. For instance, a company might use inequalities to determine the range of prices that will ensure profitability, considering production costs and market demand. Similarly, consumers can use inequalities to manage their budgets, ensuring that their spending does not exceed their income. Linear programming, a powerful optimization technique widely used in business and economics, heavily relies on the use of inequalities to define constraints and feasible regions.

2. Engineering

Engineers use inequalities to design safe and efficient structures and systems. For example, when designing a bridge, engineers need to ensure that the structure can withstand a certain range of loads and stresses. Inequalities are used to define the allowable limits for these parameters, ensuring the safety and stability of the bridge. Similarly, in electrical engineering, inequalities are used to determine the operating range of circuits and electronic components, preventing damage and ensuring proper functionality.

3. Computer Science

In computer science, inequalities are used in algorithm analysis and optimization. For example, inequalities can be used to determine the time complexity of an algorithm, which is a measure of how the algorithm's running time increases with the size of the input. This information is crucial for selecting the most efficient algorithm for a particular task. Inequalities are also used in network routing and resource allocation, ensuring that data is transmitted efficiently and resources are used effectively.

4. Everyday Life

Beyond these technical fields, inequalities also appear in everyday life. Consider situations like planning a road trip within a budget, maintaining a healthy diet by limiting calorie intake, or understanding speed limits on roads. In each of these cases, inequalities provide a framework for setting boundaries and making decisions within those boundaries. The ability to think in terms of inequalities enhances our problem-solving skills and enables us to make more informed choices in various aspects of life.

Conclusion

Solving compound inequalities like $-5 \leq \frac{4-3m}{2} < 1$ requires a methodical approach and a solid understanding of inequality properties. By carefully applying algebraic manipulations and remembering the crucial rule about flipping the inequality sign, we can accurately determine the solution set. The solution, $\frac{2}{3} < m \leq \frac{14}{3}$, represents a range of values that satisfy the original inequality. Visualizing the solution on a number line and using interval notation can enhance our understanding. Avoiding common mistakes, such as forgetting to flip the inequality sign or performing unequal operations, is essential for accurate results. Furthermore, recognizing the real-world applications of inequalities underscores their importance in various fields and everyday life. Mastering the techniques for solving inequalities not only strengthens our mathematical skills but also equips us with valuable tools for problem-solving and decision-making in a wide range of contexts.