Solving The Inequality 15 + X ≤ 24 A Step-by-Step Guide

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When tackling mathematical inequalities, understanding the fundamental principles is key to arriving at the correct solution. In this comprehensive guide, we'll delve into the process of solving the inequality 15 + x ≤ 24, providing a step-by-step explanation and ensuring a clear understanding of the concepts involved. This detailed exploration will not only provide the answer but also equip you with the knowledge to confidently solve similar problems in the future. Let's embark on this mathematical journey together!

Understanding Inequalities

Before diving into the specifics of the given inequality, it's crucial to grasp the general concept of inequalities. Unlike equations, which assert the equality of two expressions, inequalities express a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. These relationships are represented by the symbols >, <, ≥, and ≤, respectively.

In the realm of mathematics, inequalities play a pivotal role in defining ranges and constraints. Inequalities are fundamental tools for representing situations where exact equality isn't necessary or applicable. For instance, consider scenarios like speed limits on roads, where the actual speed must be less than or equal to the limit, or budget constraints, where spending should not exceed a certain amount. Understanding inequalities is crucial for problem-solving across various disciplines, including economics, engineering, and computer science. By mastering the techniques for solving inequalities, we gain the ability to model and analyze a wide range of real-world phenomena, making informed decisions and optimizing outcomes.

In the inequality 15 + x ≤ 24, the symbol '≤' signifies 'less than or equal to'. This means that the expression 15 + x must have a value that is either less than 24 or equal to 24. To solve for 'x', we need to isolate the variable on one side of the inequality, just as we would in an equation. However, there's a critical distinction to keep in mind: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. This is a crucial rule that ensures the solution set remains accurate. Ignoring this rule can lead to incorrect conclusions and a misunderstanding of the possible values of the variable.

Isolating the Variable

Our primary goal is to isolate 'x' on one side of the inequality. To achieve this, we employ inverse operations, just like when solving equations. In this case, we need to eliminate the constant term '15' from the left side of the inequality. The inverse operation of addition is subtraction, so we subtract 15 from both sides of the inequality. This maintains the balance of the relationship, ensuring that the solution set remains unchanged. By performing this operation carefully and methodically, we take a significant step towards determining the range of values that satisfy the inequality. This process of isolating the variable is a cornerstone of solving inequalities and requires a solid understanding of algebraic principles.

Performing the Subtraction

Subtracting 15 from both sides of the inequality is the key step in isolating 'x'. This operation maintains the balance of the inequality, ensuring that the relationship between the two expressions remains valid. When we subtract 15 from both sides, we effectively eliminate the constant term on the left side, bringing us closer to determining the value or range of values for 'x'.

  • Starting with the original inequality: 15 + x ≤ 24
  • Subtract 15 from both sides: (15 + x) - 15 ≤ 24 - 15
  • This simplifies to: x ≤ 9

This resulting inequality, x ≤ 9, is a significant milestone in our solution process. It tells us that the value of 'x' must be less than or equal to 9 to satisfy the original inequality. This is a precise and concise answer that directly addresses the problem's requirement. The simplicity of this solution highlights the power of algebraic manipulation in simplifying complex relationships and revealing the underlying truths. Let's further explore the implications of this solution and how it fits into the broader context of inequality problems.

The Solution

After performing the subtraction, we arrive at the solution: x ≤ 9. This inequality states that 'x' can be any value that is less than or equal to 9. This means that 9 is the upper bound of the solution set, and any number smaller than 9 is also a valid solution.

Understanding the Solution Set

The solution x ≤ 9 represents a range of values, not just a single value. To fully grasp this, it's helpful to visualize the solution set on a number line. Imagine a number line extending infinitely in both directions. The point 9 is marked on this line, and the solution includes all numbers to the left of 9, as well as 9 itself. This is often represented graphically with a closed circle at 9 (to indicate inclusion) and an arrow extending to the left, signifying all values less than 9. This visual representation reinforces the idea that there are infinitely many solutions to the inequality.

The concept of a solution set is fundamental in understanding inequalities. Unlike equations, which typically have a finite number of solutions, inequalities often have an infinite number of solutions. This is because the variable can take on any value within the specified range. Recognizing and interpreting solution sets is crucial for applying inequalities to real-world scenarios. For example, if 'x' represents the number of items a person can buy with a limited budget, the solution set would represent all the possible quantities they can afford. Therefore, mastering the concept of solution sets is essential for effectively using inequalities as a problem-solving tool.

Verification of the Solution

To ensure the correctness of our solution, it's always a good practice to verify it. This involves substituting values from the solution set back into the original inequality to see if they hold true. This step not only confirms the accuracy of our calculations but also deepens our understanding of the inequality itself.

  • Let's test a value less than 9, such as x = 5:
    • Substitute x = 5 into the original inequality: 15 + 5 ≤ 24
    • This simplifies to: 20 ≤ 24, which is true.
  • Now, let's test the boundary value, x = 9:
    • Substitute x = 9 into the original inequality: 15 + 9 ≤ 24
    • This simplifies to: 24 ≤ 24, which is also true.

These tests provide strong evidence that our solution, x ≤ 9, is correct. By verifying the solution with different values, we gain confidence in our understanding of the inequality and the steps we took to solve it. This practice of verification is a valuable habit to cultivate, as it can prevent errors and enhance problem-solving skills. Furthermore, it highlights the importance of the boundary value in inequalities, as it represents the point where the relationship transitions from true to false.

Conclusion

In conclusion, the solution to the inequality 15 + x ≤ 24 is x ≤ 9. This means that any value of 'x' that is less than or equal to 9 will satisfy the given inequality. By following the steps outlined – understanding inequalities, isolating the variable, and verifying the solution – you can confidently solve similar problems. Remember to pay close attention to the direction of the inequality sign and the importance of verifying your solutions.