Solving Absolute Value Equations: Find The Single Solution!

Hey guys! Today, we're diving into the fascinating world of absolute value equations. Specifically, we're on the hunt for the equation that gives us just one solution. Sounds like a quest, right? Let's equip ourselves with the right knowledge and get started!

Understanding Absolute Value

Before we jump into the equations, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. Because distance is always non-negative, the absolute value of a number is always non-negative. For example, |3| = 3 and |-3| = 3. Think of it as stripping away the negative sign, if there is one.

Now, when we solve absolute value equations, we often end up with two possible solutions because both a positive number and its negative counterpart will have the same absolute value. For instance, if |x| = 5, then x could be either 5 or -5. Understanding this dual possibility is crucial for spotting the equation with only one solution.

However, there's a special case: when the absolute value is equal to zero. The only number whose distance from zero is zero is zero itself! So, if we have an equation like |x| = 0, there's only one solution: x = 0. This is the key to identifying the correct answer in this problem.

Key Concepts Recap:

• Absolute Value: Distance from zero (always non-negative).
• Two Solutions: Typically, |x| = a (where a > 0) has two solutions: x = a and x = -a.
• One Solution: |x| = 0 has only one solution: x = 0.

Analyzing the Equations

Okay, armed with our understanding of absolute value, let's examine each equation to determine which one has only one solution:

$\bigcirc$ $|x-5|=-1$

Alright, let's tackle the first equation: |x - 5| = -1. Remember our golden rule? Absolute value cannot be negative! The absolute value of any expression will always be greater than or equal to zero. So, right off the bat, we can see that this equation has no solution at all. You can't take the absolute value of something and get a negative number. This is a fundamental property of absolute values, and it's super important to remember. Therefore, this isn't the equation we're looking for. We need an equation with one solution, not zero!

Think of it like this: no matter what number you plug in for x, after you subtract 5, take the absolute value, you'll never get -1. It's mathematically impossible! So, we can confidently eliminate this option.

$\bigcirc$ $|-6-2 x|=8$

Next up, we have |-6 - 2x| = 8. This equation looks a bit more promising. Since the absolute value is equal to a positive number (8), we know there could be two solutions. To find them, we need to consider both possibilities: the expression inside the absolute value could be equal to 8, or it could be equal to -8.

Let's set up two separate equations:

1. -6 - 2x = 8
2. -6 - 2x = -8

Solving the first equation:

-6 - 2x = 8

-2x = 14

x = -7

Solving the second equation:

-6 - 2x = -8

-2x = -2

x = 1

So, we have two distinct solutions: x = -7 and x = 1. Therefore, this equation has two solutions, not one. We're getting closer, but this isn't the right answer either.

$\bigcirc$ $|5 x+10|=10$

Now let's analyze the equation |5x + 10| = 10. Similar to the previous equation, since the absolute value is equal to a positive number (10), we anticipate finding two solutions. Let's break it down:

We need to consider two cases:

1. 5x + 10 = 10
2. 5x + 10 = -10

Solving the first equation:

5x + 10 = 10

5x = 0

x = 0

Solving the second equation:

5x + 10 = -10

5x = -20

x = -4

As we can see, we have two different solutions for x: x = 0 and x = -4. Thus, this equation also doesn't fit our criteria of having only one solution. Keep searching!

$\bigcirc$ $|-6 x+3|=0$

Finally, let's examine the equation |-6x + 3| = 0. Ah, this looks promising! Remember our special case? An absolute value equals zero only when the expression inside the absolute value is zero. This is our key!

So, to solve this equation, we simply need to set the expression inside the absolute value equal to zero:

-6x + 3 = 0

Now, let's solve for x:

-6x = -3

x = -3 / -6

x = 1/2

Voila! We found only one solution: x = 1/2. This is because the absolute value of zero is zero, and there's only one number that satisfies that condition. This equation perfectly matches our requirement of having only one solution.

The Verdict

After carefully analyzing each equation, we've determined that the equation with only one solution is:

$\bigcirc$ $|-6 x+3|=0$

Key Takeaways

• Remember that absolute value represents distance from zero and is always non-negative.
• Most absolute value equations (where the absolute value equals a positive number) have two solutions.
• Absolute value equations where the absolute value equals zero have only one solution.
• If an absolute value equation is set equal to a negative number, it has no solution.

So there you have it, folks! Solving absolute value equations can be a breeze once you understand the core concepts. Keep practicing, and you'll become a pro in no time! Peace out!