Math Problem: Solving Absolute Value Equations

Hey guys, let's dive into a cool math problem today! We're going to tackle an absolute value equation and figure out what values of 'x' make it true. The equation we're working with is $|3x-9|=9$. Don't let the absolute value bars scare you; they just mean we're interested in the distance of a number from zero, so the result is always non-negative. When you see an absolute value equation like this, it means that the expression inside the bars, $3x-9$, can be equal to either 9 or -9. That's the key to solving these! So, we'll set up two separate equations to find our possible solutions for 'x'.

First, let's consider the case where the expression inside the absolute value is positive. This gives us our first equation: $3x-9 = 9$. To solve for 'x' here, we want to get 'x' all by itself. We'll start by adding 9 to both sides of the equation to isolate the term with 'x'. This gives us $3x = 9 + 9$, which simplifies to $3x = 18$. Now, to get 'x' completely alone, we just need to divide both sides by 3. So, $x = 18 / 3$, and that means $x = 6$. Awesome! We've found one possible solution.

Now, let's move on to the second case. Remember, the expression inside the absolute value could also be equal to -9. So, our second equation is $3x-9 = -9$. Again, our goal is to isolate 'x'. We'll start by adding 9 to both sides of the equation. This gives us $3x = -9 + 9$, which simplifies to $3x = 0$. To find 'x', we divide both sides by 3. So, $x = 0 / 3$, and that means $x = 0$. Boom! We've found our second possible solution.

So, the solutions to the equation $|3x-9|=9$ are $x=6$ and $x=0$. If we want to check our work, we can plug these values back into the original equation. For $x=6$: $|3(6)-9| = |18-9| = |9| = 9$. It checks out! For $x=0$: $|3(0)-9| = |0-9| = |-9| = 9$. It also checks out! Both values make the original equation true. Therefore, the correct answer is d. $x=6$ and $x=0$. Keep practicing these, guys, and you'll be absolute value pros in no time!

Understanding Absolute Value Equations

Let's get a little deeper into why we split absolute value equations into two cases. When we talk about the absolute value of a number, say $|a|$, we're essentially asking 'how far is $a$ from zero on the number line?' Distance is always a positive quantity, right? So, $|5|$ is 5 because 5 is 5 units away from zero. Similarly, $|-5|$ is also 5 because -5 is also 5 units away from zero. This is why, when we have an equation like $|X| = k$ (where $k$ is a positive number), it means that $X$ must be a number that is $k$ units away from zero. There are two such numbers: $k$ itself and $-k$. This is the fundamental principle we apply when solving equations involving absolute values. In our specific problem, $|3x-9|=9$, the expression $3x-9$ is acting as our 'X'. Since the absolute value of this expression must equal 9, it means that $3x-9$ itself must be either 9 or -9. This is why we broke the problem down into the two distinct scenarios: $3x-9 = 9$ and $3x-9 = -9$. Each scenario represents a valid possibility for the expression inside the absolute value bars to yield a result of 9. It’s like saying, 'What number, when its distance from zero is taken, equals 9?' The answers are 9 and -9. So, the stuff inside our absolute value bars must be one of those two numbers.

The properties of absolute value are crucial here. For any real number $a$, $|a| less 0$ (the absolute value is never negative). If $|a| = |b|$, it doesn't necessarily mean $a=b$. It could be that $a = -b$. This is precisely what happens in absolute value equations. When we have $|expression| = positive number$, we're essentially saying that the $expression$ can be equal to that positive number OR the negative of that positive number. This understanding is key to avoiding mistakes and ensuring we find all possible solutions. If we only considered one case (e.g., $3x-9 = 9$), we would miss out on the other valid solution ($x=0$ in this case). Therefore, mastering the concept of splitting absolute value equations into two distinct cases is fundamental for accurately solving these types of algebraic problems. It’s not just a trick; it’s a direct consequence of how absolute value works mathematically. So, next time you see those bars, remember the two possibilities they represent!

Step-by-Step Solution Breakdown

Let's recap the process and make sure it's crystal clear, guys. When you're faced with an absolute value equation of the form $|expression| = positive number$, the first thing you want to do is recognize that the expression inside the absolute value can be equal to either the positive value or the negative value on the other side of the equation. This is the golden rule! So, for our problem, $|3x-9|=9$, we immediately know we need to consider two possibilities:

1. Case 1: The expression equals the positive value.

   • Here, we set up the equation: $3x - 9 = 9$.
   • Our goal is to isolate 'x'. First, we add 9 to both sides to move the constant term away from the 'x' term: $3x - 9 + 9 = 9 + 9$.
   • This simplifies to $3x = 18$.
   • Finally, we divide both sides by 3 to solve for 'x': $x = 18 / 3$.
   • This gives us our first solution: $x = 6$.

2. Case 2: The expression equals the negative value.

   • Here, we set up the second equation: $3x - 9 = -9$.
   • Again, we want to isolate 'x'. We start by adding 9 to both sides of the equation: $3x - 9 + 9 = -9 + 9$.
   • This simplifies to $3x = 0$.
   • To find 'x', we divide both sides by 3: $x = 0 / 3$.
   • This gives us our second solution: $x = 0$.

After finding these two potential solutions, $x=6$ and $x=0$, it's always a good practice to check them by substituting them back into the original equation to ensure they hold true. This verification step is super important in mathematics to confirm our work.

• Checking $x=6$:

  • Original equation: $|3x-9|=9$
  • Substitute $x=6$: $|3(6)-9| = |18-9| = |9|$.
  • Since $|9| = 9$, this solution is correct.

• Checking $x=0$:

  • Original equation: $|3x-9|=9$
  • Substitute $x=0$: $|3(0)-9| = |0-9| = |-9|$.
  • Since $|-9| = 9$, this solution is also correct.

Both values satisfy the original equation. Therefore, the complete set of solutions is $x=6$ and $x=0$. This aligns with option d from the provided choices. Remember this systematic approach, and you'll be able to solve any similar absolute value equation that comes your way, guys!

Why Other Options Are Incorrect

Let's take a moment to understand why the other options (a, b, and c) aren't the correct solutions for the equation $|3x-9|=9$. It's common to get tripped up by absolute value problems, so breaking down why the incorrect answers are wrong can be just as helpful as understanding the correct solution!

Option a. $x=6$ and x=-rac{56}{9}: We already confirmed that $x=6$ is indeed a correct solution. However, x=-rac{56}{9} is not. Let's see what happens if we try to derive this value. Perhaps there was a calculation error when solving the second case, $3x-9 = -9$. If, for instance, someone mistakenly wrote $3x-9 = -9$ and then tried to solve it by subtracting 9 from both sides (which would be incorrect, as we need to add 9), they might get something like $3x = -9 - 9$, leading to $3x = -18$, and thus $x = -6$. Or maybe they made a mistake in the first case, $3x-9=9$, where they incorrectly calculated $3x = 9-9$ (subtracting 9 instead of adding) to get $3x=0$ and $x=0$. It seems the -rac{56}{9} part might come from a completely different problem or a very specific type of arithmetic error that isn't immediately obvious from the standard way of solving this equation. The key takeaway here is that both parts of the answer must be correct for the entire option to be valid.

Option b. $x=6$: This option includes one of our correct solutions, $x=6$. However, remember that absolute value equations like this often have two solutions. By only providing $x=6$, this option implies that there isn't another value of 'x' that satisfies the equation. As we've shown through our step-by-step process, when we considered the case where $3x-9 = -9$, we found that $x=0$ is also a valid solution. Therefore, an incomplete set of solutions makes this option incorrect.

Option c. x=rac{5}{7} and $x=-1$: Neither of these values are solutions to our original equation, $|3x-9|=9$. Let's quickly test one. If we plug in x=rac{5}{7}: |3(rac{5}{7})-9| = |rac{15}{7} - rac{63}{7}| = |-rac{48}{7}| = rac{48}{7}. And rac{48}{7} is definitely not equal to 9. If we plug in $x=-1$: $|3(-1)-9| = |-3-9| = |-12| = 12$. And 12 is not equal to 9 either. These values likely come from a completely different equation, perhaps one with different coefficients or a different value on the right-hand side. It highlights the importance of carefully substituting the proposed solutions back into the original equation to verify them. Always double-check your calculations!

In summary, the only option that contains the complete and correct set of solutions derived from correctly solving the absolute value equation $|3x-9|=9$ is d. $x=6$ and $x=0$. Keep up the great work, mathematicians!