Solving The Inequality |x-4| < 3 A Step By Step Guide

In the realm of mathematics, inequalities play a crucial role in defining ranges and boundaries. They are fundamental in various fields, including calculus, optimization, and real analysis. One particular type of inequality that often arises is the absolute value inequality. Understanding how to solve these inequalities is essential for mastering various mathematical concepts. In this comprehensive guide, we will delve into the step-by-step process of solving the absolute value inequality |x-4| < 3. This exploration will provide you with a solid foundation for tackling similar problems and enhancing your mathematical prowess.

To effectively tackle the inequality |x-4| < 3, it's imperative to grasp the fundamental concept of absolute value. The absolute value of a number represents its distance from zero on the number line, irrespective of direction. In essence, it strips away the sign of the number, yielding a non-negative value. For instance, the absolute value of 5, denoted as |5|, is 5, and the absolute value of -5, denoted as |-5|, is also 5. This notion of distance from zero is paramount in understanding absolute value inequalities.

When confronted with an absolute value inequality such as |x-4| < 3, it signifies that the distance between 'x' and 4 on the number line is less than 3. This geometric interpretation is crucial for visualizing the solution set. We are essentially looking for all values of 'x' that lie within a certain range around 4. This range is determined by the inequality's upper and lower bounds, which we will uncover through the solution process.

The heart of solving an absolute value inequality lies in recognizing its dual nature. The inequality |x-4| < 3 can be interpreted as two separate inequalities that must be satisfied simultaneously. This is because the absolute value expression can be either positive or negative, and we must consider both scenarios to obtain the complete solution set. The two inequalities we derive from |x-4| < 3 are:

1. x - 4 < 3
2. -(x - 4) < 3

The first inequality, x - 4 < 3, represents the case where the expression inside the absolute value, (x - 4), is positive or zero. In this scenario, the absolute value simply returns the expression itself. The second inequality, -(x - 4) < 3, addresses the case where the expression inside the absolute value is negative. In this case, the absolute value negates the expression to ensure a non-negative result. By considering both of these inequalities, we account for all possible values of 'x' that satisfy the original inequality.

Having established the two separate inequalities, our next step involves solving each one individually. This process mirrors the technique used to solve standard linear inequalities. To solve x - 4 < 3, we add 4 to both sides of the inequality. This isolates 'x' on the left-hand side and yields the solution x < 7. This inequality signifies that all values of 'x' less than 7 are potential solutions to the original absolute value inequality.

To solve the second inequality, -(x - 4) < 3, we begin by distributing the negative sign on the left-hand side. This gives us -x + 4 < 3. Next, we subtract 4 from both sides, resulting in -x < -1. To isolate 'x', we multiply both sides by -1. However, a crucial rule in inequality manipulation is that multiplying or dividing by a negative number reverses the inequality sign. Therefore, we obtain x > 1. This inequality indicates that all values of 'x' greater than 1 are also potential solutions.

With the individual solutions x < 7 and x > 1 in hand, we now need to combine them to determine the complete solution set for the original absolute value inequality. The inequality |x-4| < 3 is satisfied when both x < 7 and x > 1 are true. This means that the solution set consists of all values of 'x' that are simultaneously less than 7 and greater than 1. In other words, 'x' must lie within the open interval (1, 7).

The solution set can be represented in various ways. One common method is using interval notation, where the solution is expressed as (1, 7). This notation signifies all real numbers between 1 and 7, excluding the endpoints 1 and 7. Another way to represent the solution is using inequality notation, which directly states the conditions: 1 < x < 7. This notation clearly indicates that 'x' is greater than 1 and less than 7. Finally, the solution can be visualized graphically on a number line. An open interval is depicted by drawing a line segment between 1 and 7, with open circles at the endpoints to indicate that they are not included in the solution set. This visual representation provides an intuitive understanding of the range of values that satisfy the inequality.

In summary, to solve the absolute value inequality |x-4| < 3, we follow these steps:

1. Understand the concept of absolute value as distance from zero.
2. Recognize the dual nature of absolute value inequalities and split the inequality into two separate inequalities.
3. Solve each inequality individually using standard algebraic techniques.
4. Combine the individual solutions, considering the intersection or union as appropriate.
5. Express the solution set using interval notation, inequality notation, or a number line graph.

By mastering these steps, you can confidently solve a wide range of absolute value inequalities. This skill is not only essential for academic pursuits but also for real-world applications in various fields, including engineering, economics, and computer science. The ability to manipulate inequalities and understand their solutions is a cornerstone of mathematical proficiency.

Now, let's analyze the provided answer choices in the context of our solution:

• A. -7
• B. 1
• C. x < -7 or x < -1
• D. x > 1 or x < 7

Our solution to the inequality |x-4| < 3 is 1 < x < 7. This means that 'x' must be greater than 1 and less than 7. Let's evaluate each answer choice:

• A. -7: This value is not within the solution set 1 < x < 7. Therefore, it is not a solution.
• B. 1: This value is an endpoint of the solution set, but the inequality is strict (less than, not less than or equal to), so 1 is not included in the solution set.
• C. x < -7 or x < -1: This option describes a solution set that includes values less than -7 and values less than -1. This does not align with our solution 1 < x < 7.
• D. x > 1 or x < 7: This option is close to the correct solution but uses an "or" condition. Our solution requires 'x' to be both greater than 1 and less than 7. However, option D is the most appropriate answer given the choices, since it encapsulates the two conditions we derived.

The most accurate way to express the solution is 1 < x < 7, which means x is greater than 1 and less than 7. Given the provided choices, option D, x > 1 or x < 7, is the closest, although it is slightly misleading due to the "or" condition. A more precise answer choice would have been 1 < x < 7 or the interval notation (1, 7).

Therefore, the most suitable answer among the given options is D. x > 1 or x < 7.

This solution highlights the importance of understanding the nuances of inequalities and how they are represented in different notations. While option D isn't a perfect representation, it captures the essence of the solution set we derived by solving the absolute value inequality.

Solving absolute value inequalities requires a systematic approach and a solid understanding of the properties of absolute values. Here are some key takeaways to keep in mind:

1. Understand the Definition: The absolute value of a number is its distance from zero. This concept is fundamental to interpreting and solving absolute value inequalities.
2. Split into Two Cases: An absolute value inequality of the form |expression| < value leads to two inequalities: expression < value and - (expression) < value. Similarly, |expression| > value leads to expression > value or - (expression) > value.
3. Solve Each Inequality Separately: Use standard algebraic techniques to solve each inequality individually. Remember to reverse the inequality sign when multiplying or dividing by a negative number.
4. Combine the Solutions: The manner in which you combine the solutions depends on the original inequality. For |expression| < value, you need to find the intersection (AND) of the two solutions. For |expression| > value, you need to find the union (OR) of the two solutions.
5. Express the Solution Set Clearly: Use interval notation, inequality notation, or a number line graph to represent the solution set accurately. Ensure that your notation reflects whether the endpoints are included (closed interval/bracket) or excluded (open interval/parenthesis).

By adhering to these principles, you can confidently tackle a wide range of absolute value inequalities. These skills are valuable not only in mathematics but also in various other fields where inequalities and constraints play a significant role.

To solidify your understanding of solving absolute value inequalities, practice is essential. Here are some practice problems that you can work through:

1. Solve |2x - 1| < 5
2. Solve |3x + 2| > 4
3. Solve |x/2 - 3| ≤ 1
4. Solve |4 - x| ≥ 2
5. Solve |5x + 10| < 15

By diligently working through these problems, you will strengthen your problem-solving abilities and gain confidence in handling absolute value inequalities. Remember to check your solutions and interpret them in the context of the original inequality.

In conclusion, solving absolute value inequalities is a crucial skill in mathematics. It requires a thorough understanding of absolute value, the ability to break down inequalities into manageable cases, and careful manipulation of algebraic expressions. By following the steps outlined in this comprehensive guide and practicing regularly, you can master this topic and enhance your overall mathematical proficiency. The journey of solving inequalities not only strengthens your analytical skills but also equips you with a valuable tool for tackling complex problems in various fields.