Solving The Inequality |(3x+4)/2| - 5 ≥ 8 A Comprehensive Guide

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Inequalities are mathematical statements that compare two expressions using symbols such as greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). Unlike equations, which have definite solutions, inequalities often have a range of solutions. Solving inequalities involves finding the set of values that satisfy the given inequality. This article aims to provide a detailed explanation of how to solve the absolute value inequality |(3x+4)/2| - 5 ≥ 8. We will break down the steps, explain the concepts, and offer additional insights to ensure a thorough understanding.

Before diving into the specific problem, it’s crucial to understand absolute value inequalities. The absolute value of a number is its distance from zero on the number line. For instance, |3| = 3 and |-3| = 3. When dealing with absolute value inequalities, we need to consider two cases: the expression inside the absolute value can be either positive or negative. This is because both positive and negative values with the same magnitude will have the same absolute value.

For an inequality of the form |x| ≥ a, where a is a positive number, the solution includes all values of x that are either greater than or equal to a, or less than or equal to -a. Mathematically, this is expressed as x ≥ a or x ≤ -a. Similarly, for an inequality of the form |x| ≤ a, the solution includes all values of x that are between -a and a, inclusive. This is expressed as -a ≤ x ≤ a.

To solve the inequality |(3x+4)/2| - 5 ≥ 8, we will follow these steps:

  1. Isolate the Absolute Value Expression: The first step is to isolate the absolute value expression on one side of the inequality. To do this, we add 5 to both sides of the inequality:

    |(3x+4)/2| - 5 + 5 ≥ 8 + 5

    |(3x+4)/2| ≥ 13

  2. Set Up Two Separate Inequalities: Next, we consider the two cases for absolute value inequalities. The expression inside the absolute value, (3x+4)/2, can be either greater than or equal to 13, or less than or equal to -13. This gives us two separate inequalities:

    Case 1: (3x+4)/2 ≥ 13

    Case 2: (3x+4)/2 ≤ -13

  3. Solve Case 1: (3x+4)/2 ≥ 13:

    To solve this inequality, we first multiply both sides by 2 to eliminate the fraction:

    2 * (3x+4)/2 ≥ 13 * 2

    3x + 4 ≥ 26

    Next, we subtract 4 from both sides:

    3x + 4 - 4 ≥ 26 - 4

    3x ≥ 22

    Finally, we divide both sides by 3:

    3x / 3 ≥ 22 / 3

    x ≥ 22/3

    So, the solution for Case 1 is x ≥ 22/3.

  4. Solve Case 2: (3x+4)/2 ≤ -13:

    Similarly, we multiply both sides by 2:

    2 * (3x+4)/2 ≤ -13 * 2

    3x + 4 ≤ -26

    Subtract 4 from both sides:

    3x + 4 - 4 ≤ -26 - 4

    3x ≤ -30

    Divide both sides by 3:

    3x / 3 ≤ -30 / 3

    x ≤ -10

    Thus, the solution for Case 2 is x ≤ -10.

  5. Combine the Solutions: The solution to the original inequality is the union of the solutions from both cases. Therefore, the solution is:

    x ≥ 22/3 or x ≤ -10

To better understand the solution, we can represent it graphically on a number line. The solution x ≥ 22/3 means all values of x that are greater than or equal to 22/3 (approximately 7.33). The solution x ≤ -10 means all values of x that are less than or equal to -10. On a number line, we would mark a closed circle at -10 and shade to the left, and mark a closed circle at 22/3 and shade to the right. This visual representation helps to solidify the concept of a range of solutions.

While the step-by-step method is straightforward, there are alternative approaches that can be used to solve absolute value inequalities. One such method involves squaring both sides of the inequality. However, this method requires careful consideration of the cases and can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed inequality but not the original inequality. Therefore, it is generally recommended to stick to the standard method of splitting the inequality into two cases.

When solving absolute value inequalities, there are several common mistakes that students often make. One common mistake is forgetting to consider both the positive and negative cases of the absolute value expression. Another mistake is incorrectly applying the rules for inequalities when multiplying or dividing by a negative number. Remember that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

For instance, if you have -2x > 4, dividing both sides by -2 gives x < -2, not x > -2. Failing to reverse the inequality sign will lead to an incorrect solution.

Additionally, students sometimes make errors in the algebraic manipulations, such as incorrectly distributing or combining like terms. It’s always a good practice to double-check each step to ensure accuracy.

Inequalities are not just abstract mathematical concepts; they have numerous practical applications in various fields. For example, in engineering, inequalities are used to define tolerance limits for the dimensions of manufactured parts. In economics, inequalities are used to model constraints and optimization problems, such as budget constraints or resource allocation. In computer science, inequalities are used in algorithms and data structures, such as sorting and searching algorithms.

Moreover, inequalities play a crucial role in statistics, where they are used to define confidence intervals and hypothesis testing. In everyday life, inequalities are used in decision-making processes, such as determining whether you have enough money to buy something or whether you have enough time to complete a task.

In conclusion, solving the inequality |(3x+4)/2| - 5 ≥ 8 involves understanding the properties of absolute value inequalities and following a systematic approach. By isolating the absolute value expression, setting up two separate inequalities, solving each case, and combining the solutions, we arrive at the final answer: x ≥ 22/3 or x ≤ -10. This comprehensive guide has not only provided a step-by-step solution but also highlighted common mistakes to avoid and the practical applications of inequalities in various fields. Mastering the techniques for solving inequalities is essential for success in mathematics and related disciplines. Understanding these concepts thoroughly will empower you to tackle more complex problems and apply mathematical reasoning in real-world scenarios.

Solving inequalities, especially those involving absolute values, might seem daunting at first. However, with a clear understanding of the underlying principles and a step-by-step approach, it becomes manageable. Remember, the key is to break down the problem into smaller, solvable parts. By mastering these skills, you’ll not only improve your mathematical abilities but also enhance your problem-solving skills in general. Whether you're a student tackling algebra or a professional applying mathematical models, a solid grasp of inequalities is invaluable.