Solving |2x - 1| - |x + 5| + 3 < 0 A Step By Step Guide

To tackle the inequality |2x - 1| - |x + 5| + 3 < 0, we need to consider different cases based on the critical points where the expressions inside the absolute value signs change their signs. These critical points are x = 1/2 and x = -5. This leads us to consider three distinct intervals: x < -5, -5 ≤ x < 1/2, and x ≥ 1/2. By analyzing each interval separately, we can eliminate the absolute value signs and solve the resulting inequalities.

1. Introduction to Absolute Value Inequalities

Absolute value inequalities can seem daunting at first, but they become manageable when approached systematically. The key is to understand that the absolute value of a number represents its distance from zero. This means that |x| can be either x (if x is non-negative) or -x (if x is negative). When dealing with inequalities involving absolute values, we must consider different cases to account for these possibilities. To solve the inequality |2x - 1| - |x + 5| + 3 < 0, the presence of absolute values necessitates a case-by-case analysis. The critical points, where the expressions inside the absolute values change signs, are x = 1/2 and x = -5. These points divide the number line into three intervals: x < -5, -5 ≤ x < 1/2, and x ≥ 1/2. By examining each interval separately, we can eliminate the absolute value signs and solve the resulting linear inequalities. This methodical approach ensures we capture all possible solutions, ultimately leading to the correct answer. Understanding the behavior of absolute values is crucial for solving inequalities that involve them. Each absolute value term requires careful consideration of its sign, which changes at specific points. In our case, |2x - 1| changes its behavior at x = 1/2, and |x + 5| changes at x = -5. These points are critical for dividing the problem into manageable intervals. In each interval, the absolute value expressions can be replaced with their equivalent expressions without absolute value signs, making the inequality easier to solve. By systematically addressing each interval, we can accurately determine the range of x values that satisfy the original inequality. This approach not only simplifies the problem but also provides a clear and logical pathway to the solution. The initial step of identifying critical points is crucial for breaking down the problem into manageable parts. These points, derived from the expressions inside the absolute values, dictate the intervals over which we need to analyze the inequality. Once the intervals are established, we can rewrite the inequality in each interval by considering the sign of the expressions within the absolute values. This transformation allows us to work with simpler, linear inequalities, making the solution process more straightforward and less prone to errors. The approach of dividing the problem into intervals is not only effective for this specific inequality but is also a valuable technique for solving a wide range of absolute value problems. It provides a structured way to handle the piecewise nature of absolute value functions, ensuring a comprehensive and accurate solution.

2. Case 1: x < -5

In this interval, both 2x - 1 and x + 5 are negative. Therefore, |2x - 1| = -(2x - 1) = -2x + 1 and |x + 5| = -(x + 5) = -x - 5. Substituting these into the inequality, we get:

(-2x + 1) - (-x - 5) + 3 < 0 -2x + 1 + x + 5 + 3 < 0 -x + 9 < 0 -x < -9 x > 9

However, this contradicts our initial assumption that x < -5. Therefore, there are no solutions in this interval. Analyzing the case where x is less than -5 is crucial for understanding the behavior of the inequality in the leftmost region of the number line. When x < -5, the expressions inside both absolute value terms, (2x - 1) and (x + 5), become negative. Consequently, the absolute values are resolved by negating the expressions. This leads to a transformed inequality that can be solved algebraically. However, it's essential to compare the solution obtained with the initial condition (x < -5) to ensure its validity. In this specific case, the solution x > 9 contradicts the initial assumption, indicating that there are no solutions within this interval. This careful comparison is a fundamental step in solving absolute value inequalities, preventing the inclusion of extraneous solutions. The lack of solutions in this interval highlights the importance of considering all possible cases when dealing with absolute values. The initial assumption dictates the form of the absolute value expressions, and any solution must adhere to this assumption. By systematically analyzing each interval, we can identify regions where the inequality holds true and regions where it does not. This rigorous approach is key to achieving an accurate and complete solution. Understanding why there are no solutions in this interval is as important as finding solutions in other intervals. It reinforces the concept of conditional solutions, where the validity of a solution depends on the initial assumptions made. This understanding is crucial for developing a deeper comprehension of absolute value inequalities and their solutions. The process of negating the expressions within the absolute values is a direct consequence of the definition of absolute value. When a quantity inside the absolute value is negative, its absolute value is obtained by multiplying it by -1, effectively changing its sign. This principle is fundamental to working with absolute values and must be applied carefully in each interval to avoid errors. The contradiction between the solution and the initial assumption serves as a clear indicator that no values of x within this interval satisfy the original inequality. This step is not merely a formality but a critical component of the problem-solving process, ensuring that only valid solutions are considered.

3. Case 2: -5 ≤ x < 1/2

In this interval, 2x - 1 is negative, and x + 5 is non-negative. Therefore, |2x - 1| = -(2x - 1) = -2x + 1 and |x + 5| = x + 5. Substituting these into the inequality, we get:

(-2x + 1) - (x + 5) + 3 < 0 -2x + 1 - x - 5 + 3 < 0 -3x - 1 < 0 -3x < 1 x > -1/3

Combining this with the interval -5 ≤ x < 1/2, we get the solution -1/3 < x < 1/2. The second case, where -5 ≤ x < 1/2, presents a different scenario where the expressions within the absolute values have opposite signs. In this interval, (2x - 1) is negative, while (x + 5) is non-negative. This difference in signs dictates how the absolute values are resolved: |2x - 1| becomes -(2x - 1), and |x + 5| remains (x + 5). Substituting these expressions into the original inequality leads to a new inequality that can be simplified and solved for x. The resulting solution, x > -1/3, must then be intersected with the initial interval -5 ≤ x < 1/2 to determine the valid solutions within this case. This intersection yields the interval -1/3 < x < 1/2, representing the range of x values that satisfy the inequality under these conditions. This process of considering the signs of the expressions within absolute values is essential for accurately solving absolute value inequalities. By carefully analyzing each case, we can ensure that the absolute values are correctly resolved, leading to valid solutions. The intersection of the solution with the initial interval is a crucial step, as it filters out any extraneous solutions that may arise from solving the simplified inequality. The interval -1/3 < x < 1/2 represents a significant portion of the solution set for the original inequality. It highlights the importance of considering all possible cases to obtain a comprehensive solution. Understanding the behavior of absolute values in different intervals is key to mastering this type of problem. The change in sign of the expressions within the absolute values dictates the form of the simplified inequality, and careful attention to these details is necessary to avoid errors. The solution obtained in this case contributes to the overall solution set of the original inequality, which will be combined with solutions from other cases to arrive at the final answer. The process of intersecting the solution with the initial interval is a fundamental technique in solving conditional inequalities. It ensures that the solutions obtained are consistent with the assumptions made at the beginning of each case, preventing the inclusion of values that do not satisfy the original inequality.

4. Case 3: x ≥ 1/2

In this interval, both 2x - 1 and x + 5 are non-negative. Therefore, |2x - 1| = 2x - 1 and |x + 5| = x + 5. Substituting these into the inequality, we get:

(2x - 1) - (x + 5) + 3 < 0 2x - 1 - x - 5 + 3 < 0 x - 3 < 0 x < 3

Combining this with the interval x ≥ 1/2, we get the solution 1/2 ≤ x < 3. The third case, where x ≥ 1/2, completes our analysis by considering the scenario where both expressions within the absolute values are non-negative. In this interval, |2x - 1| is simply 2x - 1, and |x + 5| is x + 5. Substituting these expressions into the original inequality results in a linear inequality that can be solved for x. The solution obtained, x < 3, must be considered in conjunction with the initial assumption of x ≥ 1/2. This intersection yields the interval 1/2 ≤ x < 3, representing the range of x values that satisfy the inequality under these conditions. This case highlights the importance of systematically addressing all possible intervals when solving absolute value inequalities. By considering the signs of the expressions within the absolute values, we can accurately resolve the absolute values and obtain valid solutions. The interval 1/2 ≤ x < 3 contributes another segment to the overall solution set of the original inequality. It is crucial to remember that the solution obtained in each case is conditional, meaning it is only valid within the specific interval under consideration. The process of intersecting the solution with the initial interval ensures that we only include values that are consistent with the initial assumptions. The combination of solutions from all cases will provide a comprehensive solution to the original absolute value inequality. Understanding the behavior of the expressions within the absolute values is essential for accurately solving these types of problems. By carefully considering the signs, we can simplify the inequality and obtain the correct solutions. The solution obtained in this case further illustrates the piecewise nature of absolute value functions and the importance of a case-by-case analysis. Each interval presents a different scenario, and addressing each one systematically is key to achieving a complete solution.

5. Combining the Solutions

Combining the solutions from the three cases, we have:

-1/3 < x < 1/2 from Case 2 1/2 ≤ x < 3 from Case 3

Therefore, the overall solution is -1/3 < x < 3, which can be written as the interval (-1/3, 3). To arrive at the final solution, we must combine the solutions obtained from each individual case. In Case 2, we found that -1/3 < x < 1/2, and in Case 3, we found that 1/2 ≤ x < 3. These two intervals can be combined to form a single interval that represents the complete solution set. The combination of these intervals involves considering the endpoints and determining whether they should be included or excluded. In this case, the interval from Case 2 extends up to 1/2, while the interval from Case 3 starts at 1/2. This means that 1/2 is included in the solution set. Combining these intervals, we obtain the interval -1/3 < x < 3, which represents all the x values that satisfy the original inequality. This final step of combining the solutions is crucial for providing a complete and accurate answer. It ensures that we have considered all possible cases and that we have included all valid solutions. The interval notation (-1/3, 3) is a concise way to represent the solution set, indicating that all values between -1/3 and 3, excluding -1/3 and including 3, satisfy the inequality. This final solution underscores the importance of a systematic approach to solving absolute value inequalities. By breaking the problem into cases, solving each case individually, and then combining the solutions, we can effectively tackle these types of problems. The process of combining the solutions involves careful consideration of the endpoints and the inclusion or exclusion of specific values. This step is critical for ensuring the accuracy of the final answer. The final solution, (-1/3, 3), represents the range of x values that make the original inequality true. This comprehensive solution is the culmination of the step-by-step analysis performed in each case.

6. Conclusion

Therefore, the range of values of x such that |2x - 1| - |x + 5| + 3 < 0 is (-1/3, 3), which corresponds to option (D). In conclusion, solving absolute value inequalities requires a systematic approach that involves considering different cases based on the critical points where the expressions inside the absolute value signs change their signs. By dividing the number line into intervals and analyzing each interval separately, we can eliminate the absolute value signs and solve the resulting inequalities. The solutions obtained in each interval must then be combined to determine the overall solution set. This method ensures that all possible solutions are captured and that the final answer is accurate. Absolute value inequalities can be challenging, but by following a structured approach, they can be solved effectively. The key is to understand the behavior of absolute values and to carefully consider the signs of the expressions within the absolute value signs. This allows us to rewrite the inequality in each interval without absolute values, making it easier to solve. The process of combining the solutions from each case is crucial for obtaining the complete solution set. This final step ensures that we have considered all possible scenarios and that our answer is comprehensive. The solution to the inequality |2x - 1| - |x + 5| + 3 < 0 is the interval (-1/3, 3), which means that any value of x within this interval will satisfy the inequality. This final answer is the result of a methodical analysis that involved breaking the problem into cases, solving each case individually, and then combining the solutions. The technique of case-by-case analysis is a valuable tool for solving a wide range of mathematical problems, particularly those involving absolute values. This approach allows us to address the piecewise nature of absolute value functions and to obtain accurate solutions. The final solution to the inequality, (-1/3, 3), is a testament to the power of systematic problem-solving. By carefully considering each case and combining the results, we can effectively tackle complex mathematical problems and arrive at the correct answer. This comprehensive guide provides a detailed explanation of the steps involved in solving the absolute value inequality, offering insights into the underlying concepts and techniques. By understanding these principles, readers can confidently approach similar problems and develop their problem-solving skills.