Solving Absolute Value Inequality |x| - 5 ≤ -13

Understanding Absolute Value Inequalities

When dealing with absolute value inequalities like |x| - 5 ≤ -13, 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, irrespective of direction. In other words, |x| is always non-negative. This understanding is paramount in dissecting and solving inequalities involving absolute values.

To tackle this particular inequality, we first need to isolate the absolute value term. We can achieve this by adding 5 to both sides of the inequality. This algebraic manipulation maintains the balance of the inequality and brings us closer to revealing the solution set. The transformed inequality then reads |x| ≤ -8. This step is pivotal as it sets the stage for a critical observation.

Now, considering the core definition of absolute value, we know that |x| is always greater than or equal to zero. It can never be negative. This is because distance is inherently a non-negative quantity. The absolute value represents the distance from zero, and distance cannot be a negative number. This fundamental principle is key to understanding why the given inequality has no solution.

Comparing this with our transformed inequality |x| ≤ -8, we encounter a contradiction. The absolute value |x| cannot be less than or equal to a negative number (-8 in this case). This contradiction arises from the very nature of absolute value. It highlights a critical point in solving such problems: always consider the inherent properties of mathematical functions and operations involved. In this scenario, the non-negativity of absolute value clashes directly with the requirement for it to be less than or equal to a negative number.

Therefore, based on this logical deduction, we can definitively conclude that there is no solution to the inequality |x| - 5 ≤ -13. The absolute value of any number can never be negative, making the inequality an impossibility. This conclusion underscores the importance of a solid grasp of mathematical definitions and principles when solving equations and inequalities.

Step-by-Step Solution of |x| - 5 ≤ -13

Let's dive into a step-by-step solution to the inequality |x| - 5 ≤ -13. This methodical approach will further solidify our understanding of why there's no solution. We'll break down each step, highlighting the crucial logical considerations involved.

1. Isolate the Absolute Value: The primary step involves isolating the absolute value term. We achieve this by adding 5 to both sides of the inequality: |x| - 5 + 5 ≤ -13 + 5 This simplifies to: |x| ≤ -8 This step is a standard algebraic manipulation, maintaining the integrity of the inequality while bringing the absolute value term into focus.
2. Understanding Absolute Value: As emphasized earlier, the absolute value of any real number is its distance from zero, which is always non-negative. This fundamental property of absolute value is critical for solving inequalities.
3. Analyzing the Inequality: We now have |x| ≤ -8. This inequality states that the absolute value of x is less than or equal to -8. However, we know that the absolute value of any number is always greater than or equal to zero. It can never be a negative value.
4. Identifying the Contradiction: The inequality |x| ≤ -8 presents a clear contradiction. The left side, |x|, is always non-negative, while the right side is a negative number. This contradiction indicates that there is no value of x that can satisfy this inequality.
5. Conclusion: Therefore, the inequality |x| - 5 ≤ -13 has no solution. There is no real number x whose absolute value is less than or equal to -8. This conclusion is not based on a mere algebraic manipulation; it's rooted in the fundamental definition and properties of absolute value.

In summary, by carefully isolating the absolute value term and applying the definition of absolute value, we can logically deduce that the inequality |x| - 5 ≤ -13 has no solution. This exercise highlights the significance of understanding the underlying principles of mathematical concepts, not just the mechanical steps of solving problems.

Why the Answer is No Solution

The question of why the answer to |x| - 5 ≤ -13 is “no solution” stems directly from the nature of absolute values and their inherent non-negativity. This section will delve deeper into this concept, providing a clear and intuitive explanation.

To reiterate, the absolute value of a number represents its distance from zero on the number line. This distance is always a non-negative value. Whether we're considering positive numbers, negative numbers, or zero itself, the absolute value will always be zero or a positive number. For example, |3| = 3, |-3| = 3, and |0| = 0. This non-negativity is a core characteristic of absolute values.

Now, let's revisit our inequality: |x| - 5 ≤ -13. We've already established that isolating the absolute value term leads us to |x| ≤ -8. This inequality is the crux of the matter. It states that the distance of x from zero is less than or equal to a negative number, -8. This is where the fundamental contradiction arises.

Since distance, and therefore absolute value, cannot be negative, the inequality |x| ≤ -8 is mathematically impossible. There's no real number x that can satisfy this condition. No matter what value we substitute for x, its absolute value will always be zero or positive. It can never be a negative number like -8, or any number less than -8.

Think of it this way: imagine a number line. The absolute value |x| represents the distance from zero. Can you move a certain distance away from zero and end up at a point that's less than -8 units away? No, because distance is always measured as a positive quantity. You can move 8 units to the left (reaching -8), but the distance is still 8, not -8.

This is why the answer to the inequality |x| - 5 ≤ -13 is