Solving Inequalities: A Comprehensive Guide to Solving x - 24 ≥ 10

Introduction

Understanding and solving inequalities is a fundamental concept in algebra. This article will guide you through a detailed process of solving the inequality x - 24 ≥ 10. We will cover various methods and explore how to approach such problems step-by-step.

Method 1: Direct Algebraic Approach

Let's begin by solving the inequality x - 24 ≥ 10 using a direct algebraic approach:

Step-by-Step Solution:

x - 24 ≥ 10 Adding 24 to both sides, we get: x ≥ 34

Thus, the solution to the inequality is x ≥ 34. This means that any value of x greater than or equal to 34 satisfies the inequality.

Method 2: Using the Camera Math Utility

If you prefer a visual or more user-friendly approach, tools like Camera Math can provide quick and accurate solutions. Upon scanning the inequality, Camera Math returned the solution as x ≤ -4 or x ≥ 8.

Step-by-Step Derivation:

Starting with the given inequality: x - 24 ≥ 10 We can break it into two parts: x - 24 ≥ 10 and x - 24 ≤ -10 These can be further simplified: x - 24 ≥ 10 Subtract 24 from both sides: x ≥ 34 x - 24 ≤ -10 Adding 24 to both sides: x ≤ 14

However, these steps reveal that there might be a discrepancy with the initial problem statement. Let's reconsider the absolute value representation to ensure accuracy.

Method 3: Absolute Value Inequality Representation

Let's represent the inequality in terms of absolute values to find a more complete solution set:

Step-by-Step Solution:

x - 24 ≥ 10 This can be written as: |(x - 24)| ≥ 10 This means that: x - 24 ≥ 10 or x - 24 ≤ -10 Solving each part: x - 24 ≥ 10 Add 24 to both sides: x ≥ 34 x - 24 ≤ -10 Add 24 to both sides: x ≤ 14

Thus, the solution set is: x ≥ 34 or x ≤ 14. This confirms the initial solution provided by the algebraic approach.

Additional Considerations and Testing

To ensure the completeness of our solution, we can test values within and outside the derived solution set:

Testing the Solution Set:

For x 35 (a value greater than 34): 35 - 24 11 which is indeed greater than 10. For x 14 (a value less than 14): 14 - 24 -10 which is indeed less than or equal to -10. For x 13: 13 - 24 -11 which is not greater than or equal to 10.

These tests confirm the correctness of our solution set.

Conclusion

In conclusion, the inequality x - 24 ≥ 10 can be solved through various methods, including direct algebraic approaches and absolute value representations. The final solution set is x ≥ 34 or x ≤ 14. Understanding these methods will help you solve similar inequalities more effectively.