Solving Absolute Value Inequalities $8|u-6|+5>45$

Jul 15, 2025 by ADMIN 50 views

Solving Absolute Value Inequalities: A Comprehensive Guide to $8|u-6|+5>45$

This article provides a comprehensive guide to solving the absolute value inequality $8|u-6|+5>45$ . We will break down the steps, explain the underlying concepts, and provide a clear understanding of how to arrive at the solution. Whether you're a student grappling with absolute value problems or simply looking to refresh your algebra skills, this guide will equip you with the knowledge to tackle similar inequalities with confidence.

Understanding Absolute Value

Before diving into the solution, it's crucial to grasp the concept of absolute value. The absolute value of a number is its distance from zero on the number line. This distance is always non-negative. For instance, the absolute value of 5, denoted as |5|, is 5, and the absolute value of -5, denoted as |-5|, is also 5. In essence, the absolute value strips away the sign of the number, leaving only its magnitude.

The implications of this definition are significant when dealing with equations and inequalities. When an expression involves an absolute value, it means we need to consider two possible scenarios: the expression inside the absolute value is either positive or negative. This is the core principle we'll use to solve the inequality $8|u-6|+5>45$ .

The first key concept to remember is that absolute value represents distance from zero. The absolute value of a number, denoted by two vertical bars surrounding the number (e.g., |x|), is its magnitude regardless of its sign. For example, |3| = 3 and |-3| = 3. This means that when solving inequalities involving absolute value, we need to consider two cases: one where the expression inside the absolute value is positive or zero, and another where it is negative.

Breaking Down the Inequality

In the given inequality, $8|u-6|+5>45$ , the expression inside the absolute value is (u-6). This means we need to analyze two scenarios:

When (u-6) is non-negative (u-6 ≥ 0), the absolute value |u-6| is simply (u-6).
When (u-6) is negative (u-6 < 0), the absolute value |u-6| is the opposite of (u-6), which is -(u-6).

By considering these two cases separately, we can transform the absolute value inequality into two separate linear inequalities, which are much easier to solve. This approach is fundamental to solving any absolute value inequality. Now, let's move on to the step-by-step solution of the given inequality.

Step-by-Step Solution of $8|u-6|+5>45$

Now, let's break down the solution process step-by-step. Our goal is to isolate the variable 'u' and determine the range of values that satisfy the inequality. Remember, we'll be using the principle of considering two cases based on the absolute value expression.

1. Isolate the Absolute Value Term

The first step is to isolate the absolute value term on one side of the inequality. To do this, we subtract 5 from both sides of the inequality:

$8|u-6|+5-5 > 45-5$

This simplifies to:

$8|u-6| > 40$

Next, we divide both sides by 8:

$rac{8|u-6|}{8} > rac{40}{8}$

This gives us:

$|u-6| > 5$

Now that we have isolated the absolute value term, we can proceed to the next crucial step: splitting the inequality into two separate cases.

2. Split into Two Cases

As discussed earlier, the absolute value inequality $|u-6| > 5$ translates into two distinct inequalities:

Case 1: The expression inside the absolute value is greater than 5:

$u-6 > 5$
Case 2: The expression inside the absolute value is less than -5:

$u-6 < -5$

These two cases represent the two possibilities arising from the absolute value. If the distance of (u-6) from zero is greater than 5, it means (u-6) is either greater than 5 or less than -5. We've now transformed our original absolute value inequality into two simpler linear inequalities, each representing one of these possibilities. We will now solve each of these inequalities separately.

3. Solve Each Inequality

Let's solve each inequality separately:

Case 1: $u-6 > 5$

Add 6 to both sides:

$u-6+6 > 5+6$

This gives us:

$u > 11$

So, one part of our solution is that u must be greater than 11.
Case 2: $u-6 < -5$

Add 6 to both sides:

$u-6+6 < -5+6$

This gives us:

$u < 1$

So, the other part of our solution is that u must be less than 1.

We have now found the solutions for both cases. The solutions to these inequalities will give us the complete solution set for the original absolute value inequality.

4. Combine the Solutions

The solution to the original inequality $8|u-6|+5>45$ is the union of the solutions from the two cases. This means that the values of 'u' that satisfy the inequality are either greater than 11 or less than 1. We can express this solution in several ways:

Inequality Notation: $u < 1$ or $u > 11$
Interval Notation: $(-\&infty;, 1) \cup (11, +\&infty;)$

The interval notation uses parentheses to indicate that the endpoints (1 and 11) are not included in the solution set. The symbol $\cup$ represents the union of the two intervals, meaning we combine all the numbers in both intervals to form the complete solution set. The symbols $-\&infty;$ and $+\&infty;$ represent negative and positive infinity, respectively, indicating that the solution extends indefinitely in both directions.

Therefore, the complete solution to the inequality $8|u-6|+5>45$ is all real numbers less than 1 or greater than 11. This means any value of 'u' within these ranges will satisfy the original absolute value inequality.

Graphical Representation of the Solution

A visual representation of the solution on a number line can further solidify understanding. We draw a number line and mark the points 1 and 11. Since the solutions are $u < 1$ and $u > 11$ , we use open circles at 1 and 11 to indicate that these points are not included in the solution. Then, we shade the regions to the left of 1 and to the right of 11, representing all the numbers less than 1 and all the numbers greater than 11, respectively.

This visual representation clearly shows the range of values that satisfy the inequality, providing an intuitive understanding of the solution set. It also reinforces the concept that the absolute value inequality leads to two separate regions on the number line, representing the two cases we considered during the solution process. Graphical representation is a powerful tool for understanding inequalities, especially those involving absolute value.

Common Mistakes to Avoid

When solving absolute value inequalities, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them and ensure accuracy.

Forgetting to Consider Both Cases: The most common mistake is forgetting that the expression inside the absolute value can be either positive or negative. Failing to split the inequality into two cases will lead to an incomplete or incorrect solution. Remember, the absolute value inequality requires considering both scenarios to capture the full range of solutions.
Incorrectly Handling the Negative Case: When dealing with the negative case, it's crucial to remember that the absolute value of a negative expression is its opposite. This means you need to multiply the expression inside the absolute value by -1 before proceeding. A sign error in this step can significantly alter the solution.
Incorrectly Combining Solutions: After solving the two cases, you need to combine the solutions correctly. For "greater than" absolute value inequalities (like the one we solved), the solution is typically the union of the two solution sets (i.e., "or"). For "less than" absolute value inequalities, the solution is typically the intersection of the two solution sets (i.e., "and"). Incorrectly combining the solutions will lead to a wrong answer.
Arithmetic Errors: Simple arithmetic errors can easily occur during the solution process, especially when dealing with multiple steps. Double-checking your calculations and paying close attention to signs can help prevent these errors.

By avoiding these common mistakes, you can increase your confidence and accuracy in solving absolute value inequalities.

Real-World Applications of Absolute Value Inequalities

While absolute value inequalities may seem like an abstract mathematical concept, they have practical applications in various real-world scenarios. Understanding these applications can help you appreciate the relevance of this topic.

Tolerance in Manufacturing: In manufacturing, products are often made to specific dimensions with a certain tolerance. This means that the actual dimensions can vary slightly from the target dimension, but they must fall within a certain range. Absolute value inequalities can be used to express these tolerance limits. For example, if a part is supposed to be 10 cm long with a tolerance of ±0.1 cm, the actual length (x) must satisfy the inequality |x - 10| ≤ 0.1.
Error Analysis in Science and Engineering: In scientific experiments and engineering calculations, errors are inevitable. Absolute value inequalities can be used to bound these errors. For example, if you measure a quantity to be 5.2 units with a possible error of no more than 0.05 units, the true value (y) must satisfy the inequality |y - 5.2| ≤ 0.05.
Temperature Control in HVAC Systems: Heating, ventilation, and air conditioning (HVAC) systems are designed to maintain the temperature within a certain range. Absolute value inequalities can be used to model the acceptable temperature range. For example, if a thermostat is set to 72°F with a tolerance of ±2°F, the actual temperature (t) must satisfy the inequality |t - 72| ≤ 2.
Financial Analysis: In finance, absolute value inequalities can be used to analyze deviations from expected returns or target values. For example, if an investment portfolio is expected to yield a 10% return with a maximum deviation of 3%, the actual return (r) must satisfy the inequality |r - 10| ≤ 3.

These are just a few examples of how absolute value inequalities are used in real-world applications. The ability to solve these inequalities is a valuable skill in various fields.

Conclusion

In this comprehensive guide, we have thoroughly explored how to solve the absolute value inequality $8|u-6|+5>45$ . We broke down the problem into manageable steps, explained the underlying concepts, and provided a clear and concise solution. We also discussed common mistakes to avoid and explored real-world applications of absolute value inequalities.

By understanding the principles outlined in this article, you'll be well-equipped to solve a wide range of absolute value inequalities. Remember to isolate the absolute value term, split the inequality into two cases, solve each case separately, and combine the solutions correctly. With practice and a solid understanding of the concepts, you can master absolute value inequalities and apply them effectively in various contexts.