Solving Inequalities Step-by-Step Solution For -23d + 81 ≤ -98d + 1

In the realm of algebra, solving inequalities is a fundamental skill. It allows us to determine the range of values that satisfy a given condition. In this article, we will delve into the process of solving a linear inequality, specifically $-23d + 81 \leq -98d + 1$. We will break down each step, providing a clear and concise explanation to ensure a thorough understanding. By the end of this guide, you will be equipped with the knowledge and confidence to tackle similar problems. Let's embark on this journey of mathematical exploration!

Understanding Linear Inequalities

Before we dive into the solution, it's crucial to grasp the concept of linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as less than ($<$), greater than ($>$), less than or equal to ($\leq$), or greater than or equal to ($\geq$). Unlike equations, which have a single solution or a finite set of solutions, inequalities often have an infinite range of solutions. Solving inequalities involves isolating the variable on one side of the inequality symbol to determine the range of values that satisfy the given condition.

The key to solving linear inequalities lies in understanding the properties of inequalities. Most operations that apply to equations also apply to inequalities, with one crucial exception: multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol. This is because multiplying or dividing by a negative number changes the sign of the expressions, effectively flipping the number line and the relative order of the values. For instance, if we have $2 < 4$, multiplying both sides by -1 gives us $-2 > -4$. This rule is paramount in solving inequalities accurately. Furthermore, understanding how to combine like terms and simplify expressions is essential in the process. Like terms, which are terms with the same variable raised to the same power, can be combined to simplify the inequality, making it easier to isolate the variable. By applying these principles, we can methodically solve any linear inequality and find the solution set.

Step-by-Step Solution to $-23d + 81 \leq -98d + 1$

Now, let's tackle the given inequality: $-23d + 81 \leq -98d + 1$. We will proceed step-by-step, explaining each operation to ensure clarity.

1. Combine 'd' terms on one side

The first step is to gather all the terms containing the variable 'd' on one side of the inequality. To achieve this, we can add $98d$ to both sides of the inequality. This operation will eliminate the 'd' term from the right side and consolidate it on the left side. Adding the same value to both sides maintains the balance of the inequality, ensuring that the relationship between the expressions remains valid.

$-23d + 81 + 98d \leq -98d + 1 + 98d$

This simplifies to:

$75d + 81 \leq 1$

2. Isolate the term with 'd'

Next, we need to isolate the term containing 'd'. To do this, we subtract 81 from both sides of the inequality. This operation will remove the constant term from the left side, bringing us closer to isolating the variable. Subtracting the same value from both sides preserves the inequality, ensuring that the solution set remains unchanged.

$75d + 81 - 81 \leq 1 - 81$

This simplifies to:

$75d \leq -80$

3. Solve for 'd'

Finally, to solve for 'd', we divide both sides of the inequality by 75. This operation will isolate 'd' on the left side, giving us the solution set. Since we are dividing by a positive number, the direction of the inequality symbol remains the same.

$\frac{75d}{75} \leq \frac{-80}{75}$

This simplifies to:

$d \leq -\frac{80}{75}$

4. Reduce the fraction

The final step is to reduce the fraction to its lowest terms. Both 80 and 75 are divisible by 5, so we can simplify the fraction by dividing both the numerator and the denominator by 5.

$d \leq -\frac{80 \div 5}{75 \div 5}$

This simplifies to:

$d \leq -\frac{16}{15}$

Therefore, the solution to the inequality $-23d + 81 \leq -98d + 1$ is $d \leq -\frac{16}{15}$. This means that any value of 'd' that is less than or equal to $-\frac{16}{15}$ will satisfy the original inequality.

Verifying the Solution

To ensure the accuracy of our solution, it's always a good practice to verify it. We can do this by substituting a value from the solution set back into the original inequality and checking if the inequality holds true. Let's choose a value less than $-\frac{16}{15}$, such as $d = -2$. Substituting this value into the original inequality, we get:

$-23(-2) + 81 \leq -98(-2) + 1$

Simplifying this, we get:

$46 + 81 \leq 196 + 1$

$127 \leq 197$

The inequality holds true, confirming that our solution is correct. Additionally, we can test a value outside the solution set, such as $d = 0$, to further validate our answer. Substituting this value into the original inequality, we get:

$-23(0) + 81 \leq -98(0) + 1$

Simplifying this, we get:

$81 \leq 1$

This inequality is false, as expected, since 0 is not part of the solution set. This further reinforces the correctness of our solution.

Common Mistakes to Avoid

When solving inequalities, it's essential to be aware of common mistakes that can lead to incorrect solutions. One of the most frequent errors is forgetting to reverse the direction of the inequality symbol when multiplying or dividing both sides by a negative number. This mistake can significantly alter the solution set, leading to an incorrect answer. It's crucial to remember this rule and apply it diligently whenever it's necessary.

Another common mistake is incorrectly combining like terms. For instance, adding or subtracting terms that do not have the same variable or exponent can lead to errors in simplification. It's essential to ensure that only like terms are combined to maintain the accuracy of the solution. Additionally, mistakes in arithmetic, such as incorrect addition, subtraction, multiplication, or division, can also lead to incorrect results. It's always a good practice to double-check each step to minimize the chances of making such errors.

Furthermore, failing to distribute a negative sign properly when simplifying expressions can also lead to mistakes. When a negative sign precedes a set of parentheses, it's crucial to distribute the negative sign to each term inside the parentheses. Neglecting this step can result in incorrect simplification and an inaccurate solution. By being mindful of these common mistakes and taking the necessary precautions, you can significantly improve your accuracy in solving inequalities.

Applications of Linear Inequalities

Linear inequalities are not just abstract mathematical concepts; they have numerous practical applications in various fields. In everyday life, we often encounter situations where inequalities come into play. For example, when budgeting, we might use inequalities to determine how much money we can spend on different items while staying within our financial constraints. Inequalities can also be used to determine the minimum or maximum number of items we need to purchase to qualify for a discount or meet a certain requirement. In the business world, linear inequalities are widely used in optimization problems. Businesses use inequalities to determine the optimal production levels, pricing strategies, and resource allocation to maximize profits or minimize costs. For instance, a company might use inequalities to determine the number of units of a product to manufacture to meet demand while staying within its production capacity.

In the realm of science and engineering, linear inequalities are used to model constraints and limitations in various systems. For example, in physics, inequalities can be used to describe the range of possible values for physical quantities such as velocity, acceleration, or force. In engineering, inequalities are used to design structures that can withstand certain loads or stresses, ensuring safety and stability. Furthermore, linear inequalities play a crucial role in computer science, particularly in areas such as algorithm analysis and optimization. Inequalities are used to determine the efficiency of algorithms and to optimize their performance. In machine learning, inequalities are used in the formulation of constraints for training models and ensuring their accuracy. The versatility of linear inequalities makes them an indispensable tool in various disciplines, highlighting their importance in problem-solving and decision-making.

Conclusion

In this comprehensive guide, we have meticulously walked through the process of solving the linear inequality $-23d + 81 \leq -98d + 1$. We have elucidated each step, providing clear explanations and highlighting crucial concepts. From combining like terms to isolating the variable and reducing fractions, we have covered all the necessary techniques. Moreover, we have emphasized the importance of verifying the solution and avoiding common mistakes. By understanding the principles and applying them diligently, you can confidently solve a wide range of linear inequalities.

Linear inequalities are not just theoretical constructs; they are powerful tools with practical applications in various fields. From everyday budgeting to complex optimization problems in business and engineering, inequalities provide a framework for making informed decisions and solving real-world problems. By mastering the art of solving inequalities, you equip yourself with a valuable skill that will serve you well in your academic and professional pursuits. So, embrace the challenge, practice diligently, and unlock the power of linear inequalities!