Solving The Inequality -5 <= (4-3m)/2 < 1 A Step-by-Step Guide

$$

1. Introduction to Compound Inequalities
2. Understanding the Given Inequality
3. Step-by-Step Solution
   • Isolating the Term with 'm'
   • Solving for 'm'
   • Final Solution in Inequality Form
   • Expressing the Solution in Interval Notation
4. Verification of the Solution
5. Graphical Representation of the Solution
6. Common Mistakes to Avoid
7. Real-World Applications of Inequalities
8. Conclusion

1. Introduction to Compound Inequalities

Compound inequalities are mathematical statements that combine two or more inequalities into a single statement. These inequalities often involve variables and constants, and solving them requires a clear understanding of inequality properties and algebraic manipulation. This article delves into solving a specific compound inequality, demonstrating the step-by-step process and underlying principles. Understanding compound inequalities is crucial for various fields, including mathematics, engineering, and economics, as they often arise when dealing with constraints and ranges of values. The ability to solve and interpret these inequalities accurately is a fundamental skill in mathematical problem-solving. Compound inequalities, unlike simple inequalities, give us a range of possible values rather than a single solution or an infinite set in one direction. This makes them particularly useful for modeling real-world scenarios where there are both lower and upper bounds on a variable. For instance, a manufacturing process might require a temperature to be within a certain range, or a budget might have both a minimum and a maximum spending limit. By mastering the techniques for solving compound inequalities, you enhance your ability to tackle a broader range of mathematical problems and real-world applications.

2. Understanding the Given Inequality

The given compound inequality is -5 ackslashleqslant rac{4-3m}{2} < 1. This inequality states that the expression $\frac{4-3m}{2}$ lies between -5 (inclusive) and 1 (exclusive). In simpler terms, the value of the expression is greater than or equal to -5, but strictly less than 1. To solve this inequality, we need to isolate the variable 'm' by performing algebraic operations on all parts of the inequality while maintaining the relationships between them. This understanding of compound inequalities is crucial because it dictates how we approach the problem. Unlike a simple inequality, where we might just isolate the variable on one side, a compound inequality requires us to consider multiple bounds simultaneously. The central expression, in this case, $\frac{4-3m}{2}$, needs to be manipulated in such a way that 'm' is eventually isolated in the middle, revealing the range of values that satisfy the entire compound statement. The presence of both a "less than or equal to" ($\leqslant$) and a "less than" (<) symbol indicates that one boundary is inclusive (meaning the value is included in the solution set), while the other is exclusive (meaning the value is not included). This distinction is important when expressing the final solution, particularly in interval notation or when graphing the solution set.

3. Step-by-Step Solution

To solve the compound inequality -5 ackslashleqslant rac{4-3m}{2} < 1, we will perform a series of algebraic manipulations to isolate the variable 'm'. The steps are as follows:

Isolating the Term with 'm'

The first step in isolating the term with 'm' is to eliminate the fraction. We can do this by multiplying all parts of the inequality by 2:

2 \times (-5) \leqslant 2 \times rac{4-3m}{2} < 2 \times 1

This simplifies to:

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

Next, we need to isolate the term containing 'm' (which is -3m). To do this, subtract 4 from all parts of the inequality:

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

This simplifies to:

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

Solving for 'm'

Now, to solve for 'm', we need to divide all parts of the inequality by -3. Remember that when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality signs:

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

This simplifies to:

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

It is more conventional to write the inequality with the smaller value on the left, so we rewrite it as:

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

Final Solution in Inequality Form

The final solution in inequality form is $\frac{2}{3} < m \leqslant \frac{14}{3}$. This inequality states that 'm' is greater than $\frac{2}{3}$ but less than or equal to $\frac{14}{3}$. This means that 'm' can take any value within this range, excluding $\frac{2}{3}$ but including $\frac{14}{3}$. The direction of the inequality signs is crucial in accurately representing the solution set. The strict inequality ($\<$) indicates that $\frac{2}{3}$ is not part of the solution, while the non-strict inequality ($\leqslant$) indicates that $\frac{14}{3}$ is included. This distinction is important both mathematically and in practical applications where the endpoints of a range might have significant implications. For example, if 'm' represents a physical quantity, such as temperature, exceeding the upper bound might lead to equipment damage, while falling below the lower bound might result in a process failure. Therefore, correctly interpreting and applying the inequality signs is essential for ensuring the validity and applicability of the solution.

Expressing the Solution in Interval Notation

The solution expressed in interval notation is $\left(\frac{2}{3}, \frac{14}{3}\right]$. Interval notation is a concise way to represent a set of numbers that lie within a specific range. The parentheses '(' and ')' are used to denote open intervals, meaning the endpoints are not included, while brackets '[' and ']' denote closed intervals, meaning the endpoints are included. In this case, the parenthesis on the left side of the interval, $\left(\frac{2}{3},\right.$, indicates that $\frac{2}{3}$ is not part of the solution set, which aligns with the strict inequality $m > \frac{2}{3}$. The bracket on the right side, $\left.\frac{14}{3}\right]$, indicates that $\frac{14}{3}$ is part of the solution set, corresponding to the non-strict inequality $m \leqslant \frac{14}{3}$. Interval notation provides a clear and unambiguous way to communicate the solution set, particularly in more advanced mathematical contexts. It is commonly used in calculus, analysis, and other areas where precise specification of intervals is crucial. Understanding and being able to translate between inequality notation and interval notation is a fundamental skill in mathematics, enabling effective communication and problem-solving.

4. Verification of the Solution

To verify the solution, we can pick a value within the interval $\left(\frac{2}{3}, \frac{14}{3}\right]$ and substitute it back into the original inequality. Let's choose $m = 2$, which falls within this interval since $\frac{2}{3} < 2 \leqslant \frac{14}{3}$.

Substitute $m = 2$ into the original inequality:

$-5 \leqslant \frac{4 - 3(2)}{2} < 1$

$-5 \leqslant \frac{4 - 6}{2} < 1$

$-5 \leqslant \frac{-2}{2} < 1$

$-5 \leqslant -1 < 1$

This statement is true, as -1 is indeed greater than or equal to -5 and less than 1. Therefore, our solution is likely correct.

To further ensure the solution's accuracy, it's helpful to test values outside the solution interval as well. For instance, we can test a value less than $\frac{2}{3}$, such as $m = 0$, and a value greater than $\frac{14}{3}$, such as $m = 5$.

For $m = 0$:

$-5 \leqslant \frac{4 - 3(0)}{2} < 1$

$-5 \leqslant \frac{4}{2} < 1$

$-5 \leqslant 2 < 1$

This statement is false because 2 is not less than 1.

For $m = 5$:

$-5 \leqslant \frac{4 - 3(5)}{2} < 1$

$-5 \leqslant \frac{4 - 15}{2} < 1$

$-5 \leqslant \frac{-11}{2} < 1$

$-5 \leqslant -5.5 < 1$

This statement is false because -5.5 is less than -5.

The fact that values outside the interval do not satisfy the inequality further validates our solution set, confirming that the interval $\left(\frac{2}{3}, \frac{14}{3}\right]$ accurately represents all possible values of 'm' that satisfy the original compound inequality.

5. Graphical Representation of the Solution

The graphical representation of the solution on a number line provides a visual understanding of the range of values that satisfy the inequality. To represent the solution $\frac{2}{3} < m \leqslant \frac{14}{3}$, we draw a number line and mark the points $\frac{2}{3}$ and $\frac{14}{3}$.

Since the inequality is $m > \frac{2}{3}$, we use an open circle at $\frac{2}{3}$ to indicate that this value is not included in the solution. For the inequality $m \leqslant \frac{14}{3}$, we use a closed circle (or a filled-in circle) at $\frac{14}{3}$ to show that this value is included in the solution.

We then draw a line connecting these two points, representing all the values between $\frac{2}{3}$ and $\frac{14}{3}$. The line extends from just to the right of the open circle at $\frac{2}{3}$ up to and including the closed circle at $\frac{14}{3}$. This shaded line segment visually represents the solution set, making it easy to see the range of permissible values for 'm'.

Graphically representing the solution is particularly helpful when dealing with more complex inequalities or systems of inequalities. It allows for a quick and intuitive grasp of the solution set and can aid in identifying potential errors or inconsistencies in the algebraic solution. Moreover, the graphical representation highlights the distinction between open and closed intervals, reinforcing the importance of paying attention to the inequality signs. This visual aid is a valuable tool in mathematical problem-solving and communication.

6. Common Mistakes to Avoid

When solving inequalities, there are several common mistakes to avoid to ensure accuracy:

1. Forgetting to reverse the inequality sign when multiplying or dividing by a negative number: This is a critical mistake. Remember, when you multiply or divide all parts of an inequality by a negative number, you must reverse the direction of the inequality signs. For example, if you have -2x < 4, dividing by -2 gives x > -2, not x < -2.
2. Incorrectly distributing a negative sign: When dealing with expressions inside parentheses, ensure you distribute the negative sign correctly. For example, -(x - 3) should be distributed as -x + 3, not -x - 3.
3. Not performing the same operation on all parts of the inequality: In compound inequalities, it's crucial to apply the same operation to all three parts (left side, middle, and right side) to maintain the correct relationships. For instance, if you add 5 to the middle part, you must also add 5 to the left and right sides.
4. Confusing open and closed intervals: Remember that an open circle (or parenthesis in interval notation) indicates that the endpoint is not included in the solution, while a closed circle (or bracket) indicates that it is included. Misinterpreting these symbols can lead to an incorrect solution set.
5. Incorrectly graphing the solution on a number line: Make sure you use the correct type of circle (open or closed) at the endpoints and shade the appropriate region to represent the solution set accurately. Double-check that the direction of the shading corresponds to the inequality signs.
6. Skipping steps or trying to solve the inequality too quickly: It's best to solve inequalities methodically, writing out each step clearly. This helps prevent errors and makes it easier to track your work. If you skip steps, you might inadvertently make a mistake and arrive at an incorrect solution.

By being mindful of these common pitfalls and taking the time to carefully work through each step, you can significantly improve your accuracy in solving inequalities.

7. Real-World Applications of Inequalities

Real-world applications of inequalities are abundant across various fields, demonstrating their practical significance in solving everyday problems and making informed decisions. Inequalities are used to model situations where quantities are not necessarily equal but have a defined relationship, such as one quantity being greater than, less than, or within a specific range of another.

In business and economics, inequalities are used to analyze cost, revenue, and profit scenarios. For example, a company might use inequalities to determine the minimum number of units it needs to sell to achieve a certain profit level or to set pricing strategies within a competitive market. Budget constraints are also commonly modeled using inequalities, ensuring that expenses do not exceed available funds.

In engineering and science, inequalities are used to specify tolerance levels, design constraints, and safety margins. For instance, a structural engineer might use inequalities to ensure that a bridge can withstand a certain load or that a material's properties remain within acceptable limits under varying conditions. In chemistry, inequalities can be used to define the range of concentrations for a solution to maintain its desired properties.

In computer science, inequalities are used in algorithm analysis to determine the efficiency and performance of algorithms. For example, inequalities can be used to express the upper bound on the time complexity of an algorithm or to define the conditions under which a search algorithm will find a solution within a specified time frame.

In everyday life, inequalities are used in various contexts, such as setting speed limits on roads (ensuring vehicles travel at or below a certain speed), defining age restrictions for certain activities (e.g., driving, voting, purchasing alcohol), and establishing eligibility criteria for financial aid or scholarships. They are also used in personal finance to manage budgets, calculate interest rates, and compare investment options.

The versatility of inequalities in modeling real-world situations makes them an indispensable tool in various fields. Their ability to represent constraints, ranges, and relationships between quantities allows for effective problem-solving and decision-making in a wide array of contexts.

8. Conclusion

In conclusion, solving the compound inequality $-5 \leqslant \frac{4-3m}{2} < 1$ involves a series of algebraic manipulations to isolate the variable 'm'. By multiplying, subtracting, and dividing all parts of the inequality (remembering to reverse the inequality signs when dividing by a negative number), we arrive at the solution $\frac{2}{3} < m \leqslant \frac{14}{3}$. This solution can be expressed in interval notation as $\left(\frac{2}{3}, \frac{14}{3}\right]$.

Verifying the solution by substituting values within the interval back into the original inequality, as well as values outside the interval, confirms its accuracy. Graphing the solution on a number line provides a visual representation of the range of values that satisfy the inequality.

Avoiding common mistakes, such as forgetting to reverse inequality signs or incorrectly distributing negative signs, is crucial for solving inequalities correctly. Inequalities have numerous real-world applications across various fields, highlighting their importance in practical problem-solving and decision-making.

By mastering the techniques for solving inequalities, you enhance your ability to tackle a wide range of mathematical problems and real-world scenarios. The skills and concepts discussed in this article provide a solid foundation for further exploration of mathematical topics and applications.