Solving Inequalities A Step By Step Guide To $19 \geq 4 - 5n$

Introduction

In this article, we will delve into the process of solving the linear inequality $19 \geq 4 - 5n$. Inequalities, much like equations, are fundamental concepts in mathematics and are crucial for understanding a wide range of problems in various fields such as algebra, calculus, and real-world applications. This guide will provide a step-by-step approach to solving this inequality, ensuring a clear understanding of each step involved. Solving inequalities is a critical skill in mathematics, and mastering it will empower you to tackle more complex problems. Inequalities are mathematical statements that compare two expressions using symbols such as greater than ($>$), less than ($<$), greater than or equal to ($\geq$), and less than or equal to ($\leq$). Unlike equations, which seek to find exact values that make the equation true, inequalities define a range of values that satisfy the given condition. In this detailed exploration, we will not only solve the given inequality but also discuss the underlying principles and techniques applicable to a broader range of inequality problems. Understanding these concepts is vital for anyone looking to enhance their mathematical skills and apply them effectively in various contexts. We aim to make the process of solving inequalities accessible and understandable, whether you are a student learning algebra for the first time or someone looking to refresh their knowledge. By breaking down each step and providing clear explanations, we hope to build your confidence and proficiency in handling inequalities.

Step-by-Step Solution

Step 1: Isolate the Term with the Variable

The first step in solving the inequality $19 \geq 4 - 5n$ is to isolate the term containing the variable, which in this case is $-5n$. To do this, we need to eliminate the constant term on the right side of the inequality. We can achieve this by subtracting $4$ from both sides of the inequality. This maintains the balance of the inequality, similar to how we perform operations on both sides of an equation. By subtracting $4$ from both sides, we get:

$19 - 4 \geq 4 - 5n - 4$

Simplifying this, we have:

$15 \geq -5n$

This step is crucial because it brings us closer to isolating the variable $n$. Isolating the variable is a fundamental technique in solving both equations and inequalities. It allows us to focus on the term containing the variable and perform further operations to determine the range of values that satisfy the inequality. The principle behind this step is to manipulate the inequality in a way that simplifies it without changing its solution set. By subtracting $4$ from both sides, we have effectively removed the constant term from the right side, making it easier to isolate the variable term. This process is analogous to simplifying an equation, where we aim to isolate the variable to find its value. In the context of inequalities, we are not just looking for a single value but rather a range of values that make the inequality true. Therefore, each step we take must preserve the relationship between the two sides of the inequality. Isolating the variable is a key strategy in solving any inequality, and this first step is essential for setting up the rest of the solution.

Step 2: Divide Both Sides by the Coefficient of the Variable

The next step involves isolating the variable $n$ completely. We currently have $15 \geq -5n$. To get $n$ by itself, we need to divide both sides of the inequality by the coefficient of $n$, which is $-5$. However, there is a critical rule to remember when dealing with inequalities: when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the sign of the values, and thus the relationship between the two sides of the inequality. So, dividing both sides by $-5$ and flipping the inequality sign, we get:

$\frac{15}{-5} \leq \frac{-5n}{-5}$

Simplifying this, we obtain:

$-3 \leq n$

This is equivalent to:

$n \geq -3$

The reason for reversing the inequality sign when dividing by a negative number can be understood by considering a simple example. For instance, $2 < 4$ is a true statement. If we divide both sides by $-1$ without reversing the sign, we would get $-2 < -4$, which is false. However, if we reverse the sign, we get $-2 > -4$, which is true. This illustrates why it is essential to flip the inequality sign when multiplying or dividing by a negative number. In this step, dividing by the coefficient of the variable is a standard algebraic technique to isolate the variable. The crucial detail here is the sign of the coefficient. Since we are dividing by a negative number, the reversal of the inequality sign is necessary to maintain the validity of the solution. The resulting inequality, $n \geq -3$, represents the solution set, which includes all values of $n$ that are greater than or equal to $-3$. This solution set can be visualized on a number line, where a closed circle at $-3$ indicates that $-3$ is included in the solution, and the line extends to the right, representing all numbers greater than $-3$.

Solution

Therefore, the solution to the inequality $19 \geq 4 - 5n$ is $n \geq -3$. This means that any value of $n$ that is greater than or equal to $-3$ will satisfy the original inequality. To verify this solution, we can pick a few values of $n$ that meet this condition and substitute them back into the original inequality to see if it holds true. For example, let's try $n = -3$:

$19 \geq 4 - 5(-3)$

$19 \geq 4 + 15$

$19 \geq 19$

This is true, as $19$ is equal to $19$. Now, let's try a value greater than $-3$, such as $n = 0$:

$19 \geq 4 - 5(0)$

$19 \geq 4$

This is also true, as $19$ is greater than $4$. Finally, let's try a value less than $-3$, such as $n = -4$:

$19 \geq 4 - 5(-4)$

$19 \geq 4 + 20$

$19 \geq 24$

This is false, as $19$ is not greater than or equal to $24$. These examples demonstrate that the solution $n \geq -3$ correctly represents the range of values that satisfy the original inequality. The solution to an inequality is not just a single number, but a set of numbers. In this case, the solution set includes $-3$ and all real numbers greater than $-3$. This can be visualized on a number line, which is a useful tool for understanding and representing inequalities. The solution set is a continuous interval, meaning that any number within this range will satisfy the inequality. The solution $n \geq -3$ provides a clear and concise answer to the problem. It tells us exactly which values of $n$ make the original inequality true, and this understanding is crucial for solving various mathematical problems and real-world applications.

Conclusion

In summary, we have successfully solved the inequality $19 \geq 4 - 5n$ and found the solution to be $n \geq -3$. This process involved isolating the term with the variable by subtracting $4$ from both sides, and then dividing by the coefficient of the variable, which required us to reverse the inequality sign because we were dividing by a negative number. Solving inequalities is a fundamental skill in algebra, and this example illustrates the key steps involved in solving linear inequalities. Understanding these steps allows you to tackle more complex problems with confidence. Inequalities are not just abstract mathematical concepts; they have practical applications in various fields. For instance, in economics, inequalities can be used to model budget constraints or to determine the range of prices that satisfy certain conditions. In engineering, inequalities are used to ensure that designs meet certain safety standards. In computer science, they are used in algorithm analysis and optimization. The ability to solve inequalities is therefore a valuable skill that extends beyond the classroom. The steps we have outlined in this article—isolating the variable term, dividing by the coefficient, and remembering to reverse the inequality sign when necessary—are applicable to a wide range of inequality problems. By mastering these techniques, you can confidently solve inequalities and apply them to various real-world scenarios. We hope this guide has provided a clear and comprehensive understanding of how to solve the inequality $19 \geq 4 - 5n$ and has enhanced your problem-solving skills in mathematics. Remember to practice solving different types of inequalities to further solidify your understanding and build your confidence. With practice and a solid grasp of the fundamental concepts, you will be well-equipped to handle any inequality problem that comes your way.