Solving The Inequality -18 < 2c - 10 < 0 A Step-by-Step Guide

Inequalities are a fundamental concept in mathematics, and solving inequalities is a crucial skill for various mathematical applications. Compound inequalities, like the one we're tackling today, add an extra layer of complexity. Let's dive into the step-by-step process of solving the compound inequality $-18 < 2c - 10 < 0$. This exploration will provide you with a comprehensive understanding of how to approach and solve similar problems.

Understanding Compound Inequalities

A compound inequality is essentially two inequalities combined into a single statement. In our case, $-18 < 2c - 10 < 0$ means that the expression $2c - 10$ is greater than $-18$ and less than $0$. To solve this, we need to isolate the variable c while maintaining the integrity of both inequalities. The key is to perform the same operations on all three parts of the inequality: the left side, the middle expression, and the right side. This ensures that the relationships between the values remain consistent throughout the solving process. Thinking of a compound inequality as two separate inequalities linked together by an 'and' condition is a helpful way to conceptualize the problem. This means that the solution must satisfy both inequalities simultaneously. If a value satisfies one inequality but not the other, it is not part of the solution set for the compound inequality.

Step 1: Isolate the Variable Term

The initial goal is to isolate the term containing the variable, which in this case is $2c$. To do this, we need to eliminate the constant term, which is $-10$. The standard approach is to add the additive inverse of the constant term to all parts of the inequality. In this instance, we add $+10$ to each part:

$-18 + 10 < 2c - 10 + 10 < 0 + 10$

This simplifies to:

$-8 < 2c < 10$

By adding 10 to all parts, we effectively 'moved' the -10 to the other sides of the inequalities, but with the opposite sign. This maintains the balance of the inequality and brings us closer to isolating the variable 'c'. It's crucial to remember that whatever operation is performed on one part of the compound inequality must be performed on all parts to preserve the validity of the relationship. This step highlights the importance of maintaining equality or inequality across the entire statement.

Step 2: Isolate the Variable

Now that we have $-8 < 2c < 10$, the next step is to isolate c completely. The variable c is currently being multiplied by 2. To undo this multiplication, we perform the inverse operation, which is division. We divide all parts of the inequality by 2:

$-8 / 2 < 2c / 2 < 10 / 2$

This simplifies to:

$-4 < c < 5$

Dividing all parts of the inequality by 2 effectively 'scales down' the relationship proportionally, leaving 'c' isolated in the middle. It's important to note that if we were to divide by a negative number, we would need to flip the direction of the inequality signs. However, in this case, we are dividing by a positive number, so the signs remain the same. This step demonstrates the power of inverse operations in solving inequalities and equations. By applying the correct inverse operation, we can systematically isolate the variable and determine its possible values.

Step 3: Express the Solution in Interval Notation

The inequality $-4 < c < 5$ tells us that c is greater than $-4$ and less than $5$. In other words, c lies between $-4$ and $5$, but it does not include $-4$ or $5$ themselves. We can represent this solution using interval notation. Interval notation is a concise way to express a set of numbers that fall within a specific range.

For the inequality $-4 < c < 5$, the interval notation is written as:

$c ext{ } \in ext{ } (-4, 5)$

The parentheses indicate that the endpoints, $-4$ and $5$, are not included in the solution set. If the inequality included the endpoints (e.g., $-4 \leq c \leq 5$), we would use square brackets instead ([-4, 5]). Interval notation provides a clear and unambiguous way to represent solution sets for inequalities, especially when dealing with more complex ranges of values. Understanding how to convert between inequality notation and interval notation is a crucial skill in mathematics.

Therefore, the solution to the inequality $-18 < 2c - 10 < 0$ is $c \in (-4, 5)$.

Now that we've determined the solution, let's delve deeper into what it actually means. The solution $c \in (-4, 5)$ represents all the real numbers between $-4$ and $5$, excluding $-4$ and $5$ themselves. Visualizing this on a number line can provide a clearer understanding. Imagine a number line stretching infinitely in both directions. Mark the points $-4$ and $5$ on this line. The solution set consists of all the points lying between these two marks. Since $-4$ and $5$ are not included, we would represent them with open circles on the number line. The shaded area between the open circles then represents the entire solution set.

Each number within this interval, such as -3, 0, 2, or 4.999, satisfies the original compound inequality. Conversely, any number outside this interval, such as -4, -5, 5, or 6, will not satisfy the inequality. This visual representation underscores the continuous nature of the solution set. It's not just a few isolated numbers; it's an infinite number of values that fall within the specified range. Understanding the solution set in this way can be particularly helpful when applying inequalities to real-world problems. For example, if 'c' represents a temperature, the solution tells us the range of temperatures that satisfy a given condition.

When solving inequalities, especially compound inequalities, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions.

One frequent error is forgetting to perform the same operation on all parts of the inequality. Remember, a compound inequality is a single statement, and any operation must be applied consistently across all sections to maintain the balance. Another common mistake is incorrectly applying the order of operations. Be sure to simplify each part of the inequality before attempting to isolate the variable. For instance, if there are terms to combine on one side, do that first.

A crucial point to remember is the rule for multiplying or dividing by a negative number. When you multiply or divide all parts of an inequality by a negative number, you must flip the direction of the inequality signs. Failing to do this will lead to an incorrect solution. Finally, misinterpreting the meaning of interval notation can also cause problems. Be sure to understand the difference between parentheses (which indicate exclusion of the endpoint) and square brackets (which indicate inclusion of the endpoint). By carefully avoiding these common mistakes, you can significantly improve your accuracy in solving inequalities.

Inequalities are not just abstract mathematical concepts; they have numerous applications in real-world scenarios. They are used to model situations where there is a range of possible values, rather than a single fixed value. Let's explore some examples.

In engineering, inequalities are used to determine tolerances and safety margins. For example, a bridge might be designed to withstand a certain range of weights, expressed as an inequality. In finance, inequalities can be used to represent budget constraints or investment goals. For instance, a person might have a budget that limits their spending to a certain amount, or they might aim to achieve a return on investment within a specific range.

In science, inequalities are used to model physical phenomena. For instance, the temperature in a chemical reaction might need to be maintained within a certain range for the reaction to proceed correctly. In everyday life, inequalities are used to make decisions based on constraints. For example, you might need to choose a mobile phone plan that costs less than a certain amount per month. These examples demonstrate the versatility of inequalities as a tool for modeling and solving problems in various fields. By understanding inequalities, you gain a valuable skill for analyzing and interpreting real-world situations.

To solidify your understanding of solving compound inequalities, it's essential to practice. Working through various problems will help you develop confidence and identify any areas where you might need further clarification. Here are a few practice problems you can try:

1. Solve the inequality $-10 < 3x + 2 < 14$ and express the solution in interval notation.
2. Solve the inequality $1 \leq -2y + 5 \leq 9$ and express the solution in interval notation.
3. Solve the inequality $-5 < \frac{z}{2} - 1 < 2$ and express the solution in interval notation.
4. Solve the inequality $0 \leq 4w - 8 < 12$ and express the solution in interval notation.

Remember to follow the same steps we outlined earlier: isolate the variable term, isolate the variable, and then express the solution in interval notation. Don't forget to flip the inequality signs if you multiply or divide by a negative number. Working through these problems will reinforce your understanding of the process and help you develop problem-solving skills. Checking your answers against the solutions (which you can find online or in a textbook) is a good way to verify your work and identify any mistakes. The more you practice, the more comfortable you will become with solving compound inequalities.

By mastering the techniques discussed in this guide, you'll be well-equipped to tackle a wide range of inequality problems. Remember to practice regularly and apply your knowledge to real-world scenarios to deepen your understanding. This comprehensive approach will ensure that you not only solve inequalities correctly but also appreciate their significance in mathematics and beyond. Keep practicing, and you'll become proficient in solving inequalities in no time!