Solving 2|5-2x|-3 ≤ 15 A Comprehensive Guide

In this article, we will delve into the solution of the inequality $2|5-2x|-3 ext{ ≤ } 15$. This problem falls under the domain of mathematics, specifically dealing with absolute value inequalities. Understanding how to solve such inequalities is crucial for various mathematical applications and problem-solving scenarios. We will break down the problem step by step, providing a clear and concise explanation of each step involved. Our goal is to not only provide the correct answer but also to ensure a thorough understanding of the underlying concepts. This comprehensive guide will be beneficial for students, educators, and anyone interested in enhancing their mathematical skills. Let's embark on this mathematical journey together!

Understanding Absolute Value Inequalities

Before diving into the specifics of the given inequality, it's essential to grasp the fundamental concepts of absolute value inequalities. The absolute value of a number represents its distance from zero on the number line. For example, |3| = 3 and |-3| = 3. When dealing with inequalities involving absolute values, we need to consider two scenarios: the expression inside the absolute value is positive or zero, and the expression inside the absolute value is negative.

The general approach to solving an absolute value inequality of the form |ax + b| ≤ c involves breaking it down into two separate inequalities:

1. ax + b ≤ c
2. -(ax + b) ≤ c, which is equivalent to ax + b ≥ -c

These two inequalities represent the range of values for x that satisfy the original absolute value inequality. By solving these inequalities, we can determine the solution set for x. The same principles apply to inequalities of the form |ax + b| ≥ c, where we consider ax + b ≥ c and -(ax + b) ≥ c (or ax + b ≤ -c). Understanding these principles is paramount to solving our given inequality effectively. Remember, the key is to isolate the absolute value expression and then consider both positive and negative scenarios. By mastering this technique, you'll be well-equipped to tackle a wide range of absolute value inequality problems.

Step-by-Step Solution of 2|5-2x|-3 ≤ 15

To solve the inequality $2|5-2x|-3 ext{ ≤ } 15$, we will follow a systematic approach, breaking down the problem into manageable steps. Our initial goal is to isolate the absolute value expression, which will allow us to apply the principles discussed earlier. Let's begin:

Step 1: Isolate the Absolute Value

Our first step involves isolating the term with the absolute value. We start with the given inequality:

$2|5-2x|-3 ext{ ≤ } 15$

To isolate the absolute value term, we first add 3 to both sides of the inequality:

$2|5-2x| ext{ ≤ } 18$

Next, we divide both sides by 2:

$|5-2x| ext{ ≤ } 9$

Now, we have successfully isolated the absolute value expression. This simplified form allows us to proceed with the next step, where we will consider both positive and negative scenarios of the expression within the absolute value.

Step 2: Consider Both Positive and Negative Scenarios

Now that we have the inequality in the form $|5-2x| ext{ ≤ } 9$, we need to consider two separate cases:

Case 1: The expression inside the absolute value is positive or zero

In this case, we simply remove the absolute value signs:

$5-2x ext{ ≤ } 9$

Now, we solve for x:

$-2x ext{ ≤ } 4$

Divide both sides by -2 (and remember to flip the inequality sign because we are dividing by a negative number):

$x ext{ ≥ } -2$

Case 2: The expression inside the absolute value is negative

In this case, we need to consider the negative of the expression inside the absolute value:

$-(5-2x) ext{ ≤ } 9$

This is equivalent to:

$-5+2x ext{ ≤ } 9$

Now, we solve for x:

$2x ext{ ≤ } 14$

Divide both sides by 2:

$x ext{ ≤ } 7$

By considering both positive and negative scenarios, we have obtained two inequalities: $x ext{ ≥ } -2$ and $x ext{ ≤ } 7$. These inequalities define the range of values for x that satisfy the original absolute value inequality.

Step 3: Combine the Solutions

From the previous step, we have two inequalities: $x ext{ ≥ } -2$ and $x ext{ ≤ } 7$. To find the solution set for the original inequality, we need to find the intersection of these two solution sets. In other words, we are looking for the values of x that satisfy both inequalities simultaneously.

The inequality $x ext{ ≥ } -2$ represents all values of x that are greater than or equal to -2. On a number line, this would be represented by a closed interval starting at -2 and extending to positive infinity.

The inequality $x ext{ ≤ } 7$ represents all values of x that are less than or equal to 7. On a number line, this would be represented by a closed interval starting at negative infinity and extending to 7.

To find the intersection of these two intervals, we look for the region where both inequalities are satisfied. This region is the closed interval between -2 and 7, inclusive. Therefore, the solution set for the inequality $2|5-2x|-3 ext{ ≤ } 15$ is the interval $[-2, 7]$. This means that any value of x within this interval will satisfy the original inequality. The graphical representation of these solutions on a number line further clarifies the intersection, visually confirming the solution set.

Analyzing the Answer Choices

Now that we have determined the solution set for the inequality $2|5-2x|-3 ext{ ≤ } 15$ to be $[-2, 7]$, let's analyze the given answer choices:

A. $(-\infty,-2) \cup(7, \infty)$

This represents the set of all real numbers less than -2 or greater than 7. This is not the correct solution, as we found that the solution set is a closed interval between -2 and 7.

B. $(-\infty, 1.5) \cup(7.5, \infty)$

This represents the set of all real numbers less than 1.5 or greater than 7.5. This is also incorrect, as our solution set is $[-2, 7]$.

C. $[-2,7]$

This represents the closed interval between -2 and 7, inclusive. This matches our solution set, making it the correct answer.

D. $[1.5, 7.5]$

This represents the closed interval between 1.5 and 7.5, inclusive. This is not the correct solution, as our solution set is $[-2, 7]$.

Therefore, the correct answer is C. [-2, 7]. This confirms our step-by-step solution and analysis. By carefully considering each step and comparing our result with the given options, we can confidently identify the correct answer. This process not only helps in solving the specific problem but also reinforces the understanding of absolute value inequalities and problem-solving strategies.

Conclusion: Mastering Absolute Value Inequalities

In conclusion, we have successfully solved the inequality $2|5-2x|-3 ext{ ≤ } 15$ by following a structured approach. We began by understanding the fundamental concepts of absolute value inequalities, emphasizing the importance of considering both positive and negative scenarios of the expression within the absolute value. We then systematically isolated the absolute value term, broke the inequality into two cases, solved each case separately, and finally combined the solutions to obtain the solution set $[-2, 7]$.

Throughout this process, we highlighted the significance of each step, ensuring a clear and comprehensive understanding of the solution. By analyzing the given answer choices, we confirmed that option C, $[-2, 7]$, is the correct solution. This exercise not only provides the answer to the specific problem but also enhances the understanding of absolute value inequalities and problem-solving techniques in mathematics.

Mastering absolute value inequalities is crucial for various mathematical applications, including calculus, algebra, and real analysis. The skills and techniques discussed in this article will serve as a solid foundation for tackling more complex mathematical problems. By practicing and applying these concepts, you can confidently approach and solve a wide range of inequalities, further strengthening your mathematical prowess. Remember, the key is to understand the underlying principles, break down the problem into manageable steps, and carefully analyze the results. With consistent practice and a solid understanding of the concepts, you can excel in solving absolute value inequalities and other mathematical challenges.