Solving Inequalities: Find Integer Solutions For -5 ≤ 5x < 18

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In the realm of mathematics, inequalities play a crucial role in defining the range of possible values for variables. When dealing with integers, the solutions become even more specific and require careful consideration. This article delves into the process of finding all possible integer values for a variable within a given inequality. We will explore the steps involved in solving the inequality -5 ≤ 5x < 18, providing a comprehensive understanding of the solution set.

Understanding Inequalities and Integer Solutions

Before we dive into the specific problem, let's establish a solid understanding of inequalities and integer solutions. An inequality is a mathematical statement that compares two expressions using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which have specific solutions, inequalities define a range of values that satisfy the given condition. When we restrict the solutions to integers, we are looking for whole numbers (including negative numbers and zero) that fall within the defined range.

In the inequality -5 ≤ 5x < 18, we are dealing with a compound inequality, which combines two inequalities into a single statement. The expression 5x is bounded by two values: -5 on the lower end and 18 on the upper end. The “less than or equal to” symbol (≤) indicates that -5 is included in the solution set, while the “less than” symbol (<) means that 18 is not included. Our goal is to isolate the variable x and determine all the integer values that satisfy this compound inequality.

Solving the Inequality -5 ≤ 5x < 18

To find the possible values of x, we need to isolate x in the middle of the inequality. We can achieve this by performing the same operations on all three parts of the inequality. In this case, we have the inequality -5 ≤ 5x < 18. The first step is to divide all parts of the inequality by 5. This gives us:

-5 / 5 ≤ 5x / 5 < 18 / 5

Simplifying this, we get:

-1 ≤ x < 3.6

Now we have isolated x between -1 and 3.6. However, we are looking for integer values of x. This means we need to identify all whole numbers that are greater than or equal to -1 and strictly less than 3.6.

The integers that satisfy this condition are -1, 0, 1, 2, and 3. We include -1 because the inequality states that x is greater than or equal to -1. We include 0, 1, 2 and 3 because they fall between -1 and 3.6. We do not include 4, because it is greater than 3.6.

Therefore, the possible integer values of x are -1, 0, 1, 2, and 3. These are the only whole numbers that make the original inequality true.

Detailed Step-by-Step Solution

  1. Original Inequality: -5 ≤ 5x < 18
  2. Divide by 5: To isolate x, divide all parts of the inequality by 5:
    • -5 / 5 ≤ 5x / 5 < 18 / 5
  3. Simplify:
    • -1 ≤ x < 3.6
  4. Identify Integers: Find all integers that satisfy the inequality -1 ≤ x < 3.6. These are:
    • -1, 0, 1, 2, 3

Representing the Solution

The solution set can be represented in various ways:

  • Set Notation: { -1, 0, 1, 2, 3 }
  • Number Line: A number line with closed circle at -1 and open circle at 3.6, with the segment between them shaded.

Verifying the Solutions

To ensure our solutions are correct, we can substitute each integer value back into the original inequality and check if it holds true. Let's verify each solution:

  1. x = -1: -5 ≤ 5(-1) < 18 => -5 ≤ -5 < 18 (True)
  2. x = 0: -5 ≤ 5(0) < 18 => -5 ≤ 0 < 18 (True)
  3. x = 1: -5 ≤ 5(1) < 18 => -5 ≤ 5 < 18 (True)
  4. x = 2: -5 ≤ 5(2) < 18 => -5 ≤ 10 < 18 (True)
  5. x = 3: -5 ≤ 5(3) < 18 => -5 ≤ 15 < 18 (True)

All the integer values satisfy the original inequality, confirming our solution set is correct.

Common Mistakes to Avoid

When solving inequalities, it's crucial to avoid common mistakes that can lead to incorrect solutions. Here are a few pitfalls to watch out for:

  • Dividing by a Negative Number: When dividing or multiplying an inequality by a negative number, you must reverse the direction of the inequality sign. For example, if you have -2x < 4, dividing by -2 gives x > -2.
  • Forgetting to Distribute: If there are parentheses in the inequality, ensure you distribute any multiplication or division correctly.
  • Incorrectly Identifying Integers: Pay close attention to the inequality symbols. If the inequality is strict (i.e., < or >), the endpoint is not included in the solution set. If the inequality is inclusive (i.e., ≤ or ≥), the endpoint is included.
  • Not Verifying Solutions: Always substitute your solutions back into the original inequality to check for accuracy. This can help catch errors and ensure your answer is correct.

Real-World Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have numerous real-world applications. They are used in various fields, including:

  • Economics: Determining price ranges, budget constraints, and profit margins.
  • Engineering: Designing structures, controlling systems, and setting tolerances.
  • Computer Science: Optimizing algorithms, setting performance thresholds, and defining data ranges.
  • Everyday Life: Making decisions about spending, time management, and resource allocation.

For example, an inequality can be used to represent the maximum weight capacity of an elevator or the minimum score required to pass a test. Understanding how to solve inequalities is therefore a valuable skill in many areas of life.

Conclusion: Mastering Inequalities and Integer Solutions

In conclusion, finding integer solutions to inequalities involves a systematic approach. The key steps include isolating the variable, identifying the range of possible values, and selecting the integers within that range. By understanding the properties of inequalities and avoiding common mistakes, you can confidently solve these problems. Remember to always verify your solutions and recognize the real-world applications of inequalities.

The problem we addressed, finding the possible values of x in the inequality -5 ≤ 5x < 18, illustrates the process clearly. By dividing by 5, we found that -1 ≤ x < 3.6. The integers satisfying this inequality are -1, 0, 1, 2, and 3. This comprehensive solution demonstrates the importance of careful calculation and attention to detail when working with inequalities.

By mastering the techniques discussed in this article, you can confidently tackle more complex inequality problems and apply these skills in various contexts. Whether you are solving mathematical equations, making real-world decisions, or pursuing careers in STEM fields, a strong understanding of inequalities is an invaluable asset.

Finding all integer values of x that satisfy the inequality -5 ≤ 5x < 18.

Solving Inequalities Find Integer Solutions for -5 ≤ 5x < 18