Solving Absolute Value Inequalities A Step-by-Step Guide

In the realm of mathematics, absolute value inequalities often present a unique challenge to students. These problems require a careful understanding of absolute value and how it interacts with inequalities. In this article, we will dissect the inequality $|2x-4| \leq 8$, providing a step-by-step solution and exploring the underlying concepts. This comprehensive guide aims to equip you with the skills and knowledge to confidently tackle similar problems. Understanding how to solve absolute value inequalities is a foundational skill in algebra and calculus, essential for various applications in mathematics, science, and engineering. This article not only provides a solution but also explains the logic behind each step, ensuring a thorough grasp of the subject matter. We will cover the fundamental principles of absolute value, the properties of inequalities, and how to combine these concepts to solve complex problems.

Understanding Absolute Value

To effectively solve absolute value inequalities, it's crucial to grasp the fundamental concept of absolute value. The absolute value of a number represents its distance from zero on the number line. Mathematically, the absolute value of a number x, denoted as $|x|$, is defined as:

$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$

In simpler terms, if x is non-negative (zero or positive), its absolute value is x itself. If x is negative, its absolute value is the negation of x, which makes it positive. For example, $|3| = 3$ because 3 is already positive, and $|-3| = -(-3) = 3$ because the distance of -3 from zero is 3. This understanding is crucial when dealing with inequalities because the absolute value introduces two possibilities: the expression inside the absolute value can be either positive or negative, but its distance from zero remains the same. When we encounter an absolute value inequality such as $|2x - 4| \leq 8$, we are essentially saying that the distance between $2x - 4$ and zero must be less than or equal to 8. This leads to two separate inequalities that we must solve to find the range of x that satisfies the original inequality. The geometric interpretation of absolute value as a distance also provides a visual aid for understanding the solutions. On the number line, the solutions to an absolute value inequality often represent an interval or a union of intervals, which correspond to the points that satisfy the distance condition. This perspective can be particularly helpful in visualizing the problem and checking the correctness of the solutions.

Deconstructing the Inequality $|2x-4| ">≤ 8$

Now, let's delve into the specific inequality at hand: $|2x-4| \leq 8$. This inequality states that the absolute value of the expression $2x-4$ must be less than or equal to 8. As discussed earlier, this means that the expression inside the absolute value, $2x-4$, must be within 8 units of zero on the number line. This condition gives rise to two separate inequalities that we need to consider:

1. $2x - 4 \leq 8$ (The expression $2x-4$ is less than or equal to 8)
2. $-(2x - 4) \leq 8$ (The negation of the expression $2x-4$ is less than or equal to 8)

The first inequality, $2x - 4 \leq 8$, directly reflects the case where the expression $2x - 4$ is positive or zero. The second inequality, $-(2x - 4) \leq 8$, accounts for the case where the expression $2x - 4$ is negative. In this case, we negate the expression to ensure that its absolute value is considered. Solving these two inequalities separately will give us the range of x values that satisfy the original absolute value inequality. It is crucial to recognize that both inequalities must be considered because the absolute value effectively creates two scenarios: one where the expression inside the absolute value is treated as is, and another where it is treated as its negative. The solutions to these two inequalities will define the interval or intervals on the number line that represent the complete solution set for the original absolute value inequality. The ability to break down an absolute value inequality into these two separate cases is a fundamental skill in solving these types of problems.

Solving the First Inequality: $2x - 4 ">≤ 8$

Let's tackle the first inequality: $2x - 4 \leq 8$. Our goal is to isolate x on one side of the inequality. To do this, we will follow the standard algebraic steps for solving inequalities. First, we add 4 to both sides of the inequality:

$2x - 4 + 4 \leq 8 + 4$

This simplifies to:

$2x \leq 12$

Next, we divide both sides of the inequality by 2 to isolate x:

$\frac{2x}{2} \leq \frac{12}{2}$

This gives us:

$x \leq 6$

This result tells us that all values of x that are less than or equal to 6 satisfy the first part of our absolute value inequality. This is a crucial piece of the solution, representing one boundary of the possible x values. The step-by-step process of adding 4 to both sides and then dividing by 2 demonstrates the fundamental principles of manipulating inequalities. It is important to remember that when multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed. However, in this case, we are dividing by a positive number (2), so the inequality sign remains unchanged. The solution $x \leq 6$ represents a ray on the number line, extending from 6 towards negative infinity. However, this is only one part of the solution, as we still need to consider the second inequality that arises from the absolute value.

Solving the Second Inequality: $-(2x - 4) ">≤ 8$

Now, let's solve the second inequality: $-(2x - 4) \leq 8$. This inequality arises from considering the case where the expression inside the absolute value, $2x - 4$, is negative. To solve this, we first distribute the negative sign:

$-2x + 4 \leq 8$

Next, we subtract 4 from both sides:

$-2x + 4 - 4 \leq 8 - 4$

This simplifies to:

$-2x \leq 4$

Now, we divide both sides by -2. Remember, when we divide or multiply an inequality by a negative number, we must reverse the inequality sign:

$\frac{-2x}{-2} \geq \frac{4}{-2}$

This gives us:

$x \geq -2$

This result tells us that all values of x that are greater than or equal to -2 also satisfy the original absolute value inequality. This is the second crucial piece of the solution, providing the lower bound for the possible x values. The critical step in this part of the solution is reversing the inequality sign when dividing by -2. This is a common point of error for students, so it is important to emphasize and understand why this reversal is necessary. The solution $x \geq -2$ represents another ray on the number line, extending from -2 towards positive infinity. Now, to find the complete solution to the absolute value inequality, we need to combine the solutions from both inequalities.

Combining the Solutions

We have found two inequalities that must be satisfied: $x \leq 6$ and $x \geq -2$. To find the solution to the original absolute value inequality, we need to find the values of x that satisfy both of these conditions simultaneously. In other words, we are looking for the intersection of the two solution sets.

The first inequality, $x \leq 6$, tells us that x must be less than or equal to 6. The second inequality, $x \geq -2$, tells us that x must be greater than or equal to -2. Combining these two conditions, we can express the solution as a compound inequality:

$-2 \leq x \leq 6$

This inequality states that x must be greater than or equal to -2 and less than or equal to 6. In interval notation, this solution can be written as $[-2, 6]$. This interval represents all the real numbers between -2 and 6, inclusive. On the number line, this is a closed interval, meaning that both endpoints, -2 and 6, are included in the solution set. The process of combining the solutions highlights the importance of considering both cases that arise from the absolute value. The solution set is not simply the union of the two individual solutions but rather their intersection. This is because the original absolute value inequality requires that the distance between $2x - 4$ and zero be less than or equal to 8, which implies both the upper and lower bounds on the possible values of $2x - 4$.

The Final Solution

Therefore, the solution to the inequality $|2x-4| \leq 8$ is $-2 \leq x \leq 6$, which corresponds to option D. $x \geq -2$ and $x \leq 6$. This solution represents a closed interval on the number line, bounded by -2 and 6. Any value of x within this interval will satisfy the original inequality. To verify the solution, we can pick a few values within the interval and plug them back into the original inequality. For example, if we choose $x = 0$, we get $|2(0) - 4| = |-4| = 4$, which is indeed less than or equal to 8. Similarly, if we choose $x = 6$, we get $|2(6) - 4| = |12 - 4| = |8| = 8$, which also satisfies the inequality. If we choose a value outside the interval, such as $x = 7$, we get $|2(7) - 4| = |14 - 4| = |10| = 10$, which is greater than 8 and therefore does not satisfy the inequality. This verification process provides a practical way to confirm the correctness of the solution and reinforces the understanding of the inequality.

In conclusion, solving absolute value inequalities requires a careful consideration of both positive and negative cases, as well as a solid understanding of inequality manipulation. By breaking down the problem into smaller steps and thoroughly analyzing each step, we can arrive at the correct solution and gain a deeper appreciation for the underlying mathematical concepts.

Conclusion

In this comprehensive guide, we have meticulously solved the absolute value inequality $|2x-4| \leq 8$. We began by understanding the fundamental concept of absolute value and its implications for inequalities. Then, we deconstructed the given inequality into two separate inequalities, each representing a different case arising from the absolute value. We solved each inequality step-by-step, paying close attention to the rules for manipulating inequalities, such as reversing the inequality sign when dividing by a negative number. Finally, we combined the solutions from the two inequalities to arrive at the complete solution set, which is $-2 \leq x \leq 6$. This solution represents a closed interval on the number line, indicating all values of x that satisfy the original inequality.

The process of solving absolute value inequalities involves a blend of algebraic manipulation and logical reasoning. It requires a clear understanding of the definition of absolute value and the properties of inequalities. The ability to break down a complex problem into smaller, manageable steps is crucial for success. Moreover, verifying the solution by plugging in values within and outside the solution set is an important step to ensure accuracy and reinforce understanding.

Absolute value inequalities are a fundamental topic in algebra and have applications in various areas of mathematics and beyond. Mastering this topic will not only improve your problem-solving skills but also enhance your overall mathematical understanding. By following the step-by-step approach outlined in this guide, you can confidently tackle similar problems and develop a deeper appreciation for the beauty and logic of mathematics. Remember to practice regularly and seek clarification whenever needed. With consistent effort and a solid foundation, you can excel in solving absolute value inequalities and other challenging mathematical problems.