Solving Equations: Finding The Single Solution

by SLV Team 47 views

Hey math enthusiasts! Today, we're diving into the world of equations, specifically focusing on how to spot the one with a single, solitary solution. Sounds easy, right? Well, it can be, but sometimes those absolute value signs throw us for a loop. We'll break down the given options, explain the logic behind each, and make sure you're equipped to tackle these types of problems with confidence. So, grab your pencils, and let's get started!

Decoding the Absolute Value Mystery

First off, let's chat about what the absolute value even means, ya know? Think of it like a distance from zero. No matter if a number is negative or positive, the absolute value strips away the sign and tells you how far away that number is from zero. For instance, |3| = 3 and |-3| = 3. See? Both numbers are three units away from zero. This understanding is key to tackling the problems that we will see. Because absolute value deals with distance, and distance is always non-negative. This fundamental concept influences the number of solutions an absolute value equation can have. We'll see equations that have two solutions, one solution, or even no solutions. So when we are looking for a single solution, it means that there is only one value of x that satisfies the equation.

The Golden Rule: Consider Both Positives and Negatives

When solving absolute value equations, you'll generally need to consider two scenarios: the expression inside the absolute value is positive, and the expression inside the absolute value is negative. This is because both a positive and a negative value can result in the same absolute value. For instance, in the equation |x| = 5, both x = 5 and x = -5 are valid solutions. So, when we encounter absolute value equations, we must always remember to consider both possibilities. Because the values within the absolute value could be positive or negative. The number of solutions often hinges on whether these scenarios yield different solutions, the same solution, or no solutions at all. Keep that rule in mind, alright?

Analyzing the Options: Which Equation Has Only One Solution?

Alright, let's get down to the nitty-gritty. We're going to examine each of the options provided and figure out which one has just one solution.

A. ∣x−5∣=−1|x-5|=-1

Here we go with the first equation. We have |x-5| = -1. Remember what we discussed earlier about absolute values always being non-negative? Well, the absolute value of any expression is, by definition, never negative. So, it cannot equal -1. There is no value of x that makes this equation true. This equation has no solution, so option A is not our answer. This is an important concept to grasp.

B. ∣−6−2x∣=8|-6-2 x|=8

Let's move on to option B, which is |-6 - 2x| = 8. This time, we have an absolute value equal to a positive number. Good start. Let's explore how to solve it. We have to consider two cases:

  1. -6 - 2x = 8. Let's solve for x. Add 6 to both sides, which gets us -2x = 14. Divide both sides by -2, and we find that x = -7.
  2. -6 - 2x = -8. Again, let's solve for x. Add 6 to both sides and we get -2x = -2. Dividing both sides by -2 gives us x = 1.

So, we have found two possible solutions: x = -7 and x = 1. This equation has two solutions. Because the absolute value is equal to a positive number, we will always have two solutions. So option B is also not the answer, as we're looking for an equation with only one solution.

C. ∣5x+10∣=10|5 x+10|=10

Now, let's look at option C: |5x + 10| = 10. Sounds similar to option B, doesn't it? Let's break this down into the two scenarios and see what we get.

  1. 5x + 10 = 10. Solving for x, we subtract 10 from both sides. This gives us 5x = 0. Dividing by 5, we get x = 0.
  2. 5x + 10 = -10. Subtract 10 from both sides, which gives us 5x = -20. Divide both sides by 5, and we find that x = -4.

Here, we have two different solutions as well: x = 0 and x = -4. Thus, option C is not the one we are looking for. However, remember the cases in which we have only one solution. Let's see if option D is the correct answer.

D. ∣−6x+3∣=0|-6 x+3|=0

Finally, we've arrived at option D: |-6x + 3| = 0. Now, this is interesting. The absolute value equals zero. The only way the absolute value of something can equal zero is if the expression inside the absolute value is itself zero. Makes sense, right? Let's solve for x.

So, we set -6x + 3 = 0. Subtract 3 from both sides, which gives us -6x = -3. Divide both sides by -6, and we get x = 1/2.

We found only one solution, which is x = 1/2. So, this is the equation that has only one solution! The key here is the absolute value being equal to zero, which means the expression inside must also be zero. Nice work, everyone.

Conclusion: The Winning Equation

After carefully analyzing each option, we can confidently say that option D. |-6x + 3| = 0 is the equation with only one solution. Great job sticking with me and working through each problem. Keep practicing these types of problems. Remember, absolute value questions often require you to consider both positive and negative scenarios, unless the absolute value is equal to zero, which then gives us only one answer. Keep up the excellent work, and you'll become an absolute value master in no time! Keep practicing, and you'll be acing these questions on your next test, I know it!