Solving The Inequality (4m + 2)/5 ≥ M - 4 A Step-by-Step Guide
In the realm of mathematics, inequalities play a pivotal role in defining relationships between values that are not necessarily equal. Unlike equations that assert the equality of two expressions, inequalities express relationships such as greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities is a fundamental skill with applications spanning various fields, including optimization problems, calculus, and real-world scenarios where constraints and limitations are involved. Understanding how to manipulate and solve inequalities is essential for anyone seeking a strong foundation in mathematical problem-solving.
The inequality (4m + 2)/5 ≥ m - 4 presents a classic example of a linear inequality. Linear inequalities involve variables raised to the first power and can be solved using algebraic techniques similar to those used for solving linear equations. However, there are crucial differences to keep in mind, particularly when multiplying or dividing by negative numbers, as this can affect the direction of the inequality sign. In this comprehensive guide, we will break down the steps required to solve this inequality, providing a clear and methodical approach that can be applied to similar problems. By understanding the underlying principles and techniques, you will be well-equipped to tackle a wide range of inequalities with confidence.
Before diving into the solution of the specific inequality, let's establish a firm understanding of the foundational concepts related to inequalities. An inequality is a mathematical statement that compares two expressions using inequality symbols. The most common inequality symbols include:
- > (greater than)
- < (less than)
- ≥ (greater than or equal to)
- ≤ (less than or equal to)
These symbols dictate the relationship between the expressions on either side. For instance, a > b means that a is greater than b, while a ≤ b means that a is less than or equal to b. Understanding these symbols is the first step in comprehending and manipulating inequalities.
Solving an inequality involves finding the set of values that satisfy the inequality. This set of values is known as the solution set. Unlike equations, which typically have a finite number of solutions (or no solution), inequalities often have an infinite number of solutions. These solutions can be represented graphically on a number line or expressed in interval notation. For example, the inequality x > 3 represents all values of x that are greater than 3, which can be visualized as an open interval on the number line extending from 3 to infinity. The ability to interpret and represent solution sets is a critical component of working with inequalities.
Let's now delve into the step-by-step solution of the inequality (4m + 2)/5 ≥ m - 4. This process will illustrate the algebraic techniques used to isolate the variable and determine the solution set.
1. Eliminate the Fraction
The first step in solving this inequality is to eliminate the fraction. The fraction can be removed by multiplying both sides of the inequality by the denominator, which in this case is 5. Multiplying both sides by 5, we get:
5 * [(4m + 2)/5] ≥ 5 * (m - 4)
This simplifies to:
4m + 2 ≥ 5(m - 4)
By multiplying both sides by the denominator, we have successfully removed the fraction, making the inequality easier to manipulate algebraically. This is a standard technique in solving inequalities and equations involving fractions.
2. Distribute on the Right Side
Next, we need to distribute the 5 on the right side of the inequality. This involves multiplying 5 by each term inside the parentheses:
4m + 2 ≥ 5 * m - 5 * 4
This simplifies to:
4m + 2 ≥ 5m - 20
Distributing terms is a fundamental algebraic operation that ensures each term within the parentheses is properly accounted for. In this case, distributing the 5 allows us to remove the parentheses and further simplify the inequality.
3. Move Variables to One Side
To isolate the variable m, we need to move all terms containing m to one side of the inequality. A common strategy is to move the terms to the side where the coefficient of m will be positive. In this case, we can subtract 4m from both sides:
4m + 2 - 4m ≥ 5m - 20 - 4m
This simplifies to:
2 ≥ m - 20
By subtracting 4m from both sides, we have grouped the m terms on the right side, which will lead to a positive coefficient for m when we further isolate it.
4. Isolate the Variable
Now, we need to isolate m by adding 20 to both sides of the inequality:
2 + 20 ≥ m - 20 + 20
This simplifies to:
22 ≥ m
By adding 20 to both sides, we have successfully isolated m on the right side. The inequality now reads 22 ≥ m, which is equivalent to m ≤ 22. This means that the solution set includes all values of m that are less than or equal to 22.
5. Express the Solution
Finally, we express the solution in a clear and concise manner. The inequality 22 ≥ m can be rewritten as m ≤ 22. This inequality states that m is less than or equal to 22.
Solution Set:
The solution set for the inequality (4m + 2)/5 ≥ m - 4 is all values of m that are less than or equal to 22. This can be represented in several ways:
- Inequality Notation: m ≤ 22
- Interval Notation: (-∞, 22]
- Graphical Representation: On a number line, this would be a closed interval starting at negative infinity and extending to 22, with a closed circle at 22 to indicate that 22 is included in the solution set.
While the steps outlined above provide a clear method for solving linear inequalities, there are a few critical considerations to keep in mind:
1. Multiplying or Dividing by a Negative Number
One of the most important rules to remember when solving inequalities is that multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the order of the numbers on the number line. For example, if we have the inequality -2x < 4, dividing both sides by -2 would require us to flip the inequality sign, resulting in x > -2.
2. Checking the Solution
After solving an inequality, it is always a good practice to check the solution by substituting a value from the solution set back into the original inequality. This helps ensure that the solution is correct and that no errors were made during the algebraic manipulations. For example, if we found that x > 3, we could substitute x = 4 into the original inequality to verify that it holds true.
3. Compound Inequalities
Some inequalities are compound, meaning they involve two or more inequalities combined into a single statement. Compound inequalities can take the form of "and" inequalities (e.g., 2 < x < 5) or "or" inequalities (e.g., x < 2 or x > 5). Solving compound inequalities requires considering each part of the inequality separately and then combining the solutions appropriately. For "and" inequalities, the solution set is the intersection of the solutions to each inequality, while for "or" inequalities, the solution set is the union of the solutions.
The ability to solve inequalities is not just a theoretical skill; it has numerous practical applications in various fields.
1. Optimization Problems
In optimization problems, inequalities are used to define constraints and limitations. For example, a company might want to maximize its profit subject to constraints on production capacity, raw material availability, and budget. These constraints can be expressed as inequalities, and solving these inequalities helps determine the feasible region within which the optimal solution lies.
2. Real-World Scenarios
Inequalities are also used to model real-world scenarios involving comparisons and restrictions. For instance, a speed limit on a road is an inequality that restricts the maximum speed a vehicle can travel. Similarly, a budget constraint limits the amount of money that can be spent. Understanding how to work with inequalities allows us to make informed decisions in these situations.
3. Calculus
In calculus, inequalities are used to define intervals of increase and decrease for functions, determine the concavity of curves, and solve optimization problems. The ability to solve inequalities is essential for understanding the behavior of functions and for applying calculus to real-world problems.
Solving the inequality (4m + 2)/5 ≥ m - 4 demonstrates the fundamental principles and techniques involved in working with inequalities. By following a systematic approach, eliminating fractions, distributing terms, and isolating the variable, we can determine the solution set and express it in various notations. Remembering the key considerations, such as reversing the inequality sign when multiplying or dividing by a negative number, is crucial for accuracy.
The ability to solve inequalities is a valuable skill with wide-ranging applications in mathematics and beyond. Whether you are tackling optimization problems, modeling real-world scenarios, or delving into calculus, a solid understanding of inequalities will serve you well. By mastering the techniques outlined in this guide, you will be well-prepared to solve a variety of inequalities and apply them to solve real-world problems.
Solving inequalities, Mathematics, Linear inequalities, Solution set, Algebraic techniques, Optimization problems, Real-world scenarios
