Solving The Inequality 0 <= (4-3m)/2 < 1 A Step-by-Step Guide

This article delves into the intricacies of solving the inequality $-i \leqslant \frac{4-3m}{2} < 1$. This mathematical problem requires a careful step-by-step approach to isolate the variable m and determine the range of values that satisfy the given condition. Understanding the process of solving such inequalities is crucial for various applications in mathematics, physics, and engineering. Let's embark on a journey to unravel the solution to this intriguing inequality.

Understanding the Basics of Inequalities

Before diving into the specifics of the problem, it's essential to grasp the fundamental concepts of inequalities. Inequalities, unlike equations, do not provide a single solution but rather a range of solutions. They express a relationship where one value is either greater than, less than, greater than or equal to, or less than or equal to another value. The symbols used to represent these relationships are >, <, ≥, and ≤, respectively.

Solving inequalities involves manipulating the expressions on both sides to isolate the variable of interest. The golden rule to remember is that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. This rule is paramount in ensuring the solution set remains accurate. Furthermore, understanding interval notation and how to represent solutions graphically on a number line are vital skills for interpreting and communicating the solution effectively.

Step-by-Step Solution of the Inequality

Now, let's dissect the given inequality: $-i \leqslant \frac{4-3m}{2} < 1$. The presence of i might initially appear perplexing, but it's crucial to recognize that this problem likely intends for the left side of the inequality to be a numerical value, most probably 0. So we will consider the inequality $0 \leqslant \frac{4-3m}{2} < 1$. If i represents the imaginary unit, the inequality becomes significantly more complex and falls outside the scope of typical real-number inequality solutions.

Assuming the intended inequality is $0 \leqslant \frac{4-3m}{2} < 1$, we proceed as follows:

Step 1: Multiplying by 2

To eliminate the fraction, we multiply all parts of the inequality by 2. This operation maintains the inequality's direction since 2 is a positive number:

$2 * 0 \leqslant 2 * \frac{4-3m}{2} < 2 * 1$

This simplifies to:

$0 \leqslant 4 - 3m < 2$

Step 2: Isolating the Term with m

Next, we aim to isolate the term containing m. We subtract 4 from all parts of the inequality:

$0 - 4 \leqslant 4 - 3m - 4 < 2 - 4$

This results in:

$-4 \leqslant -3m < -2$

Step 3: Dividing by -3 (and Reversing Inequality Signs)

Now, we divide all parts of the inequality by -3. Remember the crucial rule: dividing by a negative number reverses the direction of the inequality signs:

$\frac{-4}{-3} \geqslant \frac{-3m}{-3} > \frac{-2}{-3}$

This simplifies to:

$\frac{4}{3} \geqslant m > \frac{2}{3}$

Step 4: Rewriting the Inequality

For clarity and conventional notation, we rewrite the inequality with m on the left and in ascending order:

$\frac{2}{3} < m \leqslant \frac{4}{3}$

Interpreting the Solution

The solution $\frac{2}{3} < m \leqslant \frac{4}{3}$ signifies that m can take any value greater than $\frac{2}{3}$ but less than or equal to $\frac{4}{3}$. In interval notation, this solution is expressed as $(\frac{2}{3}, \frac{4}{3}]$. The parenthesis indicates that $\frac{2}{3}$ is not included in the solution set, while the square bracket indicates that $\frac{4}{3}$ is included.

Graphical Representation

On a number line, this solution is represented by an open circle at $\frac{2}{3}$ and a closed circle at $\frac{4}{3}$, with a line connecting the two. The open circle signifies that $\frac{2}{3}$ is not part of the solution, and the closed circle indicates that $\frac{4}{3}$ is included. This visual representation provides a clear understanding of the range of values that satisfy the inequality.

Common Mistakes and Pitfalls

When solving inequalities, several common mistakes can lead to incorrect solutions. One of the most frequent errors is forgetting to reverse the inequality signs when multiplying or dividing by a negative number. This mistake can drastically alter the solution set. Another pitfall is misinterpreting the inequality symbols, particularly the difference between strict inequalities (>, <) and inclusive inequalities (≥, ≤). A strict inequality excludes the endpoint, while an inclusive inequality includes it.

Careless arithmetic errors can also derail the solution process. It's crucial to double-check each step to ensure accuracy. Additionally, when dealing with compound inequalities, it's essential to maintain the correct order of operations and apply the same operation to all parts of the inequality. By being mindful of these potential pitfalls, you can enhance your accuracy and confidence in solving inequalities.

The Significance of Inequalities in Mathematics and Beyond

Inequalities are not merely abstract mathematical concepts; they are powerful tools with widespread applications in various fields. In mathematics, inequalities are fundamental to calculus, analysis, and optimization problems. They are used to define intervals, establish bounds, and prove theorems. In physics, inequalities are used to describe constraints on physical quantities, such as energy and momentum. They also play a crucial role in thermodynamics and fluid dynamics.

In engineering, inequalities are used in design optimization, control systems, and resource allocation. They help engineers ensure that systems operate within safe and efficient limits. In economics, inequalities are used to model market equilibrium, consumer behavior, and resource scarcity. They provide insights into how resources are distributed and how decisions are made under constraints.

The ability to solve and interpret inequalities is a valuable skill that transcends academic boundaries. It empowers individuals to make informed decisions, solve real-world problems, and navigate complex situations where constraints and limitations are present.

Conclusion

Solving the inequality $-i \leqslant \frac{4-3m}{2} < 1$ (or more precisely, $0 \leqslant \frac{4-3m}{2} < 1$) demonstrates the systematic approach required to tackle such problems. By understanding the fundamental principles of inequalities, applying the correct algebraic manipulations, and interpreting the solution accurately, we can confidently determine the range of values that satisfy the given condition. Remember the importance of reversing inequality signs when multiplying or dividing by a negative number and the significance of interval notation and graphical representation in communicating the solution effectively. Inequalities are not just mathematical exercises; they are essential tools for problem-solving in diverse fields, making their mastery a valuable asset.