Solving Linear Inequalities A Step By Step Guide To -32c + 12 ≤ -66c - 16

Introduction

In the realm of mathematics, linear inequalities play a crucial role in defining relationships between variables and constants. Solving linear inequalities involves finding the range of values for a variable that satisfies a given inequality. This article delves into the step-by-step process of solving the linear inequality $-32c + 12 \leq -66c - 16$, providing a clear and comprehensive guide for students and enthusiasts alike. Understanding how to manipulate and simplify inequalities is fundamental in various mathematical applications, including optimization problems, calculus, and real-world scenarios. This guide aims not only to solve the given inequality but also to enhance your understanding of the underlying principles and techniques involved in handling such mathematical expressions. By mastering these concepts, you will be better equipped to tackle more complex problems and appreciate the elegance and utility of linear inequalities in mathematics and beyond.

Understanding Linear Inequalities

Before we dive into the solution, it's essential to grasp the basics of linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as $<$, $>$, $\leq$, or $\geq$. Unlike linear equations, which have a single solution, linear inequalities typically have a range of solutions. This range represents all the values that, when substituted for the variable, make the inequality true. The key to solving linear inequalities lies in applying algebraic operations while adhering to specific rules that ensure the inequality remains valid. For instance, multiplying or dividing both sides of an inequality by a negative number requires flipping the inequality sign. This article will guide you through these rules and demonstrate how to apply them effectively to solve the given inequality. Understanding these fundamental concepts is crucial for success in algebra and calculus, as inequalities are frequently encountered in problem-solving scenarios and mathematical proofs. This section lays the groundwork for the step-by-step solution that follows, ensuring you have a solid foundation in the principles of linear inequalities.

Step-by-Step Solution of $-32c + 12 \leq -66c - 16$

1. Combine Like Terms

The first step in solving the inequality is to gather the variable terms on one side and the constant terms on the other. We start with the given inequality: $-32c + 12 \leq -66c - 16$. To isolate the variable terms, we add $66c$ to both sides of the inequality. This yields: $-32c + 66c + 12 \leq -66c + 66c - 16$, which simplifies to $34c + 12 \leq -16$. Next, we want to isolate the constant terms. To do this, we subtract 12 from both sides: $34c + 12 - 12 \leq -16 - 12$. This simplifies to $34c \leq -28$. Combining like terms is a fundamental step in solving any algebraic equation or inequality. It helps to simplify the expression, making it easier to isolate the variable and find the solution. In this case, by adding $66c$ and subtracting 12, we have successfully grouped the variable and constant terms, paving the way for the next step in the solution process.

2. Isolate the Variable

Now that we have simplified the inequality to $34c \leq -28$, the next step is to isolate the variable $c$. To do this, we need to divide both sides of the inequality by the coefficient of $c$, which is 34. So, we divide both sides by 34: $\frac{34c}{34} \leq \frac{-28}{34}$. This simplifies to $c \leq \frac{-28}{34}$. Isolating the variable is a critical step in solving inequalities. It allows us to determine the range of values for the variable that satisfy the inequality. In this case, by dividing both sides by 34, we are one step closer to finding the solution for $c$. The next step will involve simplifying the fraction to its lowest terms to obtain the final solution.

3. Simplify the Fraction

The inequality is now in the form $c \leq \frac{-28}{34}$. To express the solution in its simplest form, we need to reduce the fraction $\frac{-28}{34}$ to its lowest terms. Both the numerator (-28) and the denominator (34) are divisible by 2. Dividing both by 2, we get: $\frac{-28 \div 2}{34 \div 2} = \frac{-14}{17}$. Therefore, the simplified inequality is $c \leq \frac{-14}{17}$. Reducing fractions to their lowest terms is an essential skill in mathematics. It ensures that the solution is presented in the most concise and understandable form. In this case, by dividing both the numerator and denominator by their greatest common divisor (2), we have obtained the simplified fraction $\frac{-14}{17}$, which represents the final value for the upper bound of $c$.

Final Solution

After simplifying the fraction, we arrive at the final solution for the inequality $-32c + 12 \leq -66c - 16$. The solution is: $c \leq \frac{-14}{17}$. This means that any value of $c$ that is less than or equal to $-\frac{14}{17}$ will satisfy the original inequality. It is important to present the solution in its simplest form, which we have achieved by reducing the fraction to its lowest terms. Understanding how to solve linear inequalities is a fundamental skill in algebra, with applications in various fields, including calculus, optimization, and real-world problem-solving. This solution demonstrates the step-by-step process of combining like terms, isolating the variable, and simplifying fractions to arrive at the final answer.

Verification of the Solution

To ensure the accuracy of our solution, it's crucial to verify the result. We can do this by substituting a value less than or equal to $-\frac{14}{17}$ into the original inequality and checking if it holds true. Let's choose $c = -1$ as a test value since $-1$ is less than $-\frac{14}{17}$ (approximately -0.82). Substituting $c = -1$ into the original inequality $-32c + 12 \leq -66c - 16$, we get: $-32(-1) + 12 \leq -66(-1) - 16$. This simplifies to $32 + 12 \leq 66 - 16$, which further simplifies to $44 \leq 50$. Since $44$ is indeed less than or equal to $50$, the inequality holds true for $c = -1$. This verification step provides confidence in the correctness of our solution. It's always a good practice to verify your solutions, especially in mathematics, to avoid errors and ensure a thorough understanding of the problem-solving process.

Common Mistakes to Avoid

When solving linear inequalities, several common mistakes can lead to incorrect solutions. One frequent error is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. For example, if you have $-2x < 4$, dividing both sides by -2 requires flipping the inequality sign to get $x > -2$. Another common mistake is incorrectly combining like terms. It's crucial to ensure that you are only adding or subtracting terms that have the same variable and exponent. Additionally, errors can occur when simplifying fractions. Always remember to reduce fractions to their lowest terms by dividing both the numerator and denominator by their greatest common divisor. By being aware of these common pitfalls, you can improve your accuracy and confidence in solving linear inequalities. This section serves as a reminder to pay close attention to the rules and details of algebraic manipulation, ensuring you avoid these common mistakes and arrive at the correct solution.

Conclusion

In this comprehensive guide, we have thoroughly explored the process of solving the linear inequality $-32c + 12 \leq -66c - 16$. We began by understanding the fundamentals of linear inequalities and then proceeded step-by-step through the solution, which involved combining like terms, isolating the variable, and simplifying the fraction. The final solution, $c \leq \frac{-14}{17}$, provides a clear understanding of the range of values for $c$ that satisfy the inequality. Furthermore, we verified the solution by substituting a test value and discussed common mistakes to avoid. Mastering the techniques for solving linear inequalities is essential for success in various mathematical contexts, from basic algebra to advanced calculus. This guide serves as a valuable resource for students and anyone interested in enhancing their mathematical skills. By following these steps and understanding the underlying principles, you can confidently tackle a wide range of inequality problems and appreciate the power and versatility of algebraic methods.