Solutions For |d-5| = -69? Let's Find Out!

by Mei Lin 43 views

Hey everyone! Let's dive into a fun math problem today. We're going to explore the equation |d-5| = -69 and figure out how many solutions it has. This might seem tricky at first, but I promise, it's quite straightforward once we break it down. So, grab your thinking caps, and let's get started!

Understanding Absolute Value

Before we tackle the equation directly, it's crucial to have a solid understanding of absolute value. Guys, remember that the absolute value of a number is its distance from zero on the number line. Distance is always a non-negative value. Whether you go 5 steps to the right (positive 5) or 5 steps to the left (negative 5), you've still traveled a distance of 5. Mathematically, we denote absolute value using vertical bars. For example, |5| = 5, and |-5| = 5. This means that the absolute value of any number, whether positive or negative, is always positive or zero.

Absolute value can be formally defined as follows:

|x| = x, if x ≥ 0 |x| = -x, if x < 0

This definition simply means that if the number inside the absolute value bars is positive or zero, the absolute value is the number itself. If the number inside is negative, the absolute value is the negation of that number, which makes it positive. For instance, let's consider a few examples:

  • |10| = 10 (because 10 is positive)
  • |0| = 0 (because 0 is zero)
  • |-7| = -(-7) = 7 (because -7 is negative, and its negation is 7)
  • |-3.14| = -(-3.14) = 3.14 (because -3.14 is negative, and its negation is 3.14)

Now that we have a firm grasp of what absolute value means, we can proceed to analyze the given equation with a clearer perspective. Always remember, the core concept is that absolute value represents distance, and distance cannot be negative. This understanding is the key to solving our problem.

Analyzing the Equation |d-5| = -69

Okay, now let's focus on the equation |d-5| = -69. Here's where our understanding of absolute value becomes super important. Remember, the absolute value of any expression will always be non-negative (that is, either positive or zero). It can never be a negative number. Think about it – distance can't be negative!

So, let's break this down. The left side of the equation, |d-5|, represents the absolute value of the expression 'd-5'. No matter what value 'd' takes, the absolute value of 'd-5' will always be either zero or a positive number. It will never be negative.

Now, look at the right side of the equation: -69. This is a negative number. We have a situation where the absolute value of something (|d-5|) is claimed to be equal to a negative number (-69). This contradicts the fundamental property of absolute values. An absolute value can never be negative!

This contradiction tells us something crucial: there is no value of 'd' that can make this equation true. No matter what number we substitute for 'd', the absolute value of 'd-5' will always be non-negative, and thus, it can never equal -69.

Let’s try a few examples to illustrate this point. Suppose d = 0. Then |d-5| = |0-5| = |-5| = 5, which is not -69. Suppose d = 5. Then |d-5| = |5-5| = |0| = 0, which is also not -69. Suppose d = 10. Then |d-5| = |10-5| = |5| = 5, which is still not -69. You see, no matter what value we choose for 'd', the result of |d-5| will never be -69.

This understanding is critical because it highlights the core principle of absolute values. When you see an equation where an absolute value expression is set equal to a negative number, you can immediately conclude that there are no solutions. There’s no need to do any further calculations or manipulations. The contradiction is evident right away. This can save you a lot of time and effort on tests and assignments!

The Solution: No Solutions

Given our exploration of absolute value and the equation |d-5| = -69, we've arrived at a clear conclusion: the equation has no solutions. This is because the absolute value of any expression can never be negative, but the equation tries to equate it to -69, which is a negative number. This fundamental contradiction means there's no value of 'd' that can satisfy the equation.

In mathematical terms, we say that the solution set is empty, often denoted by the symbol ∅. This symbol signifies that there are no elements (in this case, no values of 'd') that belong to the solution set. So, when you encounter a situation like this, you can confidently state that there are no solutions.

This concept is extremely important in mathematics. It highlights the significance of understanding the properties and constraints of mathematical operations and functions. Absolute value is just one example. There are other mathematical concepts, like square roots, logarithms, and trigonometric functions, that also have specific domains and ranges, which can lead to equations with no solutions if those constraints are violated.

Therefore, the key takeaway here is not just that this particular equation has no solutions, but also that you should always be mindful of the underlying principles of the mathematical operations involved. Checking for potential contradictions like an absolute value equaling a negative number is a crucial step in solving equations accurately and efficiently. So, always keep this in mind as you tackle more math problems!

Why This Matters

You might be wondering,