Solving Linear Inequalities An Example With -32c + 12 ≤ -66c - 16

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In this article, we will delve into the process of solving linear inequalities, specifically focusing on isolating the variable c. Linear inequalities are mathematical expressions that compare two values using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike linear equations, which have a single solution, linear inequalities typically have a range of solutions. This comprehensive guide will walk you through the steps involved in solving the inequality -32c + 12 ≤ -66c - 16, providing clear explanations and practical tips along the way. Our goal is to equip you with the knowledge and skills necessary to tackle similar problems with confidence.

Understanding Linear Inequalities

Before we dive into the solution, it's essential to grasp the fundamental concepts of linear inequalities. A linear inequality is a mathematical statement that compares two expressions using one of the inequality symbols mentioned earlier. These expressions involve variables raised to the power of one, which is why they are called linear. Solving a linear inequality means finding all the values of the variable that make the inequality true. The solution is often expressed as a range of values rather than a single value. When working with inequalities, there's a crucial rule to remember: multiplying or dividing both sides of the inequality by a negative number reverses the direction of the inequality symbol. This is a critical step that ensures the solution remains accurate. Understanding this rule is paramount to successfully solving linear inequalities. In the following sections, we will apply these principles to solve the given inequality.

Step-by-Step Solution

Let's tackle the inequality -32c + 12 ≤ -66c - 16 step by step:

1. Combine Like Terms

The first step in solving any inequality is to simplify the expression by combining like terms. In this case, we want to gather all the terms containing the variable c on one side of the inequality and the constant terms on the other side. To do this, we can add 66c to both sides of the inequality:

-32c + 12 + 66c ≤ -66c - 16 + 66c

This simplifies to:

34c + 12 ≤ -16

Next, we subtract 12 from both sides to isolate the term with c:

34c + 12 - 12 ≤ -16 - 12

Which gives us:

34c ≤ -28

2. Isolate the Variable

Now that we have the term with c isolated, we need to get c by itself. To do this, we divide both sides of the inequality by 34:

(34c) / 34 ≤ -28 / 34

This results in:

c ≤ -28/34

3. Simplify the Fraction

The final step is to simplify the fraction to its lowest terms. Both -28 and 34 are divisible by 2, so we can reduce the fraction:

c ≤ (-28 ÷ 2) / (34 ÷ 2)

Which simplifies to:

c ≤ -14/17

Therefore, the solution to the inequality is c ≤ -14/17. This means that any value of c that is less than or equal to -14/17 will satisfy the original inequality.

Verification of the Solution

To ensure our solution is correct, it's always a good practice to verify it. We can do this by substituting a value within the solution range and a value outside the range back into the original inequality. If the inequality holds true for a value within the range and is false for a value outside the range, we can be confident in our solution.

1. Choose a Value Within the Solution Range

Let's choose c = -1, which is less than -14/17 (approximately -0.82). Substitute this value into the original inequality:

-32*(-1)* + 12 ≤ -66*(-1)* - 16

32 + 12 ≤ 66 - 16

44 ≤ 50

This statement is true, so c = -1 satisfies the inequality.

2. Choose a Value Outside the Solution Range

Now, let's choose c = 0, which is greater than -14/17. Substitute this value into the original inequality:

-32*(0)* + 12 ≤ -66*(0)* - 16

0 + 12 ≤ 0 - 16

12 ≤ -16

This statement is false, so c = 0 does not satisfy the inequality. This verification process confirms that our solution, c ≤ -14/17, is correct.

Common Mistakes to Avoid

When solving linear inequalities, it's crucial to be aware of common mistakes that can lead to incorrect solutions. One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. As we mentioned earlier, this is a fundamental rule that must be applied to maintain the accuracy of the solution. Another common mistake is incorrectly combining like terms or making arithmetic errors during the simplification process. To avoid these errors, it's helpful to double-check each step and pay close attention to signs and operations. Additionally, failing to simplify the fraction to its lowest terms can also be considered an error, as it doesn't provide the most concise and accurate representation of the solution. By being mindful of these common pitfalls and practicing diligent problem-solving techniques, you can minimize the chances of making mistakes and arrive at the correct solution.

Real-World Applications of Linear Inequalities

Linear inequalities are not just abstract mathematical concepts; they have numerous practical applications in various real-world scenarios. Understanding how to solve them can be incredibly useful in decision-making and problem-solving across different fields. For example, in finance, linear inequalities can be used to determine investment strategies that meet specific financial goals, such as maximizing returns while staying within a certain risk tolerance. In business, they can help optimize production processes, ensuring that costs are minimized while meeting demand. In everyday life, linear inequalities can be applied to budgeting, helping individuals make informed decisions about spending and saving. Whether it's determining the maximum number of items you can purchase within a budget or calculating the minimum amount of time you need to work to earn a certain amount of money, linear inequalities provide a valuable tool for analyzing and solving real-world problems. The ability to translate real-world situations into mathematical inequalities and solve them effectively is a crucial skill in many areas of life.

Conclusion

In this article, we've provided a comprehensive guide to solving the linear inequality -32c + 12 ≤ -66c - 16. We've covered the essential steps, including combining like terms, isolating the variable, and simplifying the fraction. We also emphasized the importance of verifying the solution and highlighted common mistakes to avoid. Furthermore, we explored the real-world applications of linear inequalities, demonstrating their relevance in various fields. By understanding these concepts and practicing regularly, you can develop the skills necessary to confidently solve linear inequalities and apply them to practical problems. Remember, the key to mastering linear inequalities lies in understanding the underlying principles, paying attention to detail, and consistently practicing problem-solving techniques.