Solving Inequalities Step-by-Step Guide To Finding X

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What number goes in the green box in the inequality 3 + x/5 ≤ 8 after subtracting 3 from both sides?

In this comprehensive guide, we will walk you through the process of solving inequalities, focusing on the specific example provided. Our goal is to not only find the value of x that satisfies the inequality but also to understand the underlying principles involved. We will break down each step, ensuring clarity and a solid understanding of inequality solutions. This article is designed to help students, educators, and anyone looking to refresh their math skills.

Understanding Inequalities

Before we dive into the specific problem, it's essential to grasp what inequalities are. Inequalities, in mathematics, are used to compare values that are not necessarily equal. Unlike equations, which state that two expressions are equal, inequalities indicate that one value is greater than, less than, greater than or equal to, or less than or equal to another value. The symbols used in inequalities are:

  • < (less than)

  • (greater than)

  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)

When solving inequalities, our aim is to isolate the variable (in this case, x) on one side of the inequality sign, much like solving equations. However, there's a crucial difference: when we multiply or divide both sides of an inequality by a negative number, we must reverse the direction of the inequality sign. This is a critical rule to remember to ensure accurate solutions.

Problem Statement

The inequality we aim to solve is:

3 + x/5 ≤ 8

Our objective is to find the number that goes in the green box after performing the initial step of subtracting 3 from both sides of the inequality.

Step-by-Step Solution

Step 1: Isolate the Term with x

The first step in solving this inequality is to isolate the term containing x. In this case, the term is x/5. To isolate it, we need to eliminate the constant term, which is 3, on the left side of the inequality. We do this by subtracting 3 from both sides of the inequality. This maintains the balance of the inequality, just like we do in equations.

3 + x/5 ≤ 8
- 3       - 3

Step 2: Perform the Subtraction

Now, let's perform the subtraction. On the left side, 3 - 3 equals 0, effectively removing the constant term. On the right side, we subtract 3 from 8. This is a simple arithmetic operation, but it's crucial for moving towards our solution. The subtraction step is vital in isolating the term with the variable, setting the stage for the next steps in solving the inequality. By focusing on this subtraction, we maintain the balance of the inequality while simplifying the expression.

Step 3: Determining the Value for the Green Box

After subtracting 3 from both sides, we get:

x/5 ≤ 8 - 3
x/5 ≤ 5

So, the number that goes in the green box is 5. This step showcases the result of the subtraction, which is a key component in solving the inequality. The value 5 represents the upper limit for x/5, and it's essential for understanding the possible values of x. By correctly calculating this value, we ensure that our subsequent steps will lead to the correct solution for the inequality.

Step 4: Isolate x

Now that we have x/5 ≤ 5, our next goal is to isolate x completely. To do this, we need to eliminate the division by 5. The inverse operation of division is multiplication, so we will multiply both sides of the inequality by 5. This step is critical for solving for x and requires careful attention to maintain the integrity of the inequality.

Step 5: Multiply Both Sides by 5

Multiplying both sides of the inequality by 5, we get:

5 * (x/5) ≤ 5 * 5

This simplifies to:

x ≤ 25

This step demonstrates the multiplication operation that isolates x. By multiplying both sides by 5, we eliminate the fraction and obtain a clearer understanding of the possible values for x. The result, x ≤ 25, is a crucial milestone in solving the inequality.

Step 6: The Solution

The solution to the inequality is x ≤ 25. This means that x can be any number less than or equal to 25. This solution set includes a range of values, highlighting a key difference between inequalities and equations, where solutions are typically single values. The inequality x ≤ 25 provides a comprehensive understanding of the possible values for x that satisfy the original inequality.

Importance of Understanding Inequalities

Understanding inequalities is crucial for various reasons. They are used extensively in mathematics, science, and engineering to model real-world situations where quantities are not exactly equal. For example, inequalities can be used to describe constraints in optimization problems, such as finding the maximum profit within a certain budget or the minimum cost while meeting specific requirements. Inequalities also form the basis for more advanced mathematical concepts, such as calculus and real analysis. A solid grasp of inequalities is therefore essential for students pursuing further studies in STEM fields.

Common Mistakes to Avoid

When working with inequalities, there are a few common mistakes that students often make. One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This is a critical rule, and overlooking it can lead to incorrect solutions. Another common mistake is treating inequalities like equations, without considering the range of possible solutions. Inequalities typically have a range of solutions, rather than a single value, and it's important to represent the solution set correctly.

Tips for Solving Inequalities

To improve your skills in solving inequalities, here are some useful tips:

  1. Always remember to reverse the inequality sign when multiplying or dividing by a negative number.
  2. Treat inequalities like equations when performing addition and subtraction.
  3. Check your solution by plugging in values from the solution set into the original inequality to ensure they hold true.
  4. Graph the solution on a number line to visualize the range of possible values.
  5. Practice regularly to build confidence and fluency in solving inequalities.

Real-World Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have numerous real-world applications. In economics, inequalities are used to model supply and demand, budget constraints, and resource allocation. In physics, they are used to describe ranges of motion, forces, and energies. In computer science, inequalities are used in algorithms for optimization, sorting, and searching. Understanding inequalities allows us to model and analyze a wide variety of situations where quantities are not fixed but vary within certain limits.

Conclusion

In this article, we have explored the step-by-step solution to the inequality 3 + x/5 ≤ 8. We identified that the number that goes in the green box after the initial subtraction step is 5. We then proceeded to solve the inequality completely, finding that x ≤ 25. We also discussed the importance of understanding inequalities, common mistakes to avoid, tips for solving them, and their real-world applications. By mastering the concepts and techniques presented here, you can confidently tackle a wide range of inequality problems. Remember, practice is key to success in mathematics, so keep solving problems and building your skills!

Solving inequalities is a fundamental skill in mathematics with wide-ranging applications. By understanding the principles and following the steps outlined in this guide, you can confidently tackle inequality problems and apply them to real-world situations. Remember to pay attention to the details, avoid common mistakes, and practice regularly to build your skills. With dedication and perseverance, you can master inequalities and enhance your mathematical abilities.