Solving The Equation: 2(x-3) + 2 = 3(x-1) + 3

Alright, let's dive into solving this equation step-by-step! Our main goal here is to find the solution set, which basically means figuring out what value(s) of x will make the equation true. Don't worry, it's not as intimidating as it sounds. We'll break it down together. So, grab your favorite beverage, and let's get started!

Step 1: Distribute the Numbers

The first thing we need to do is distribute the numbers outside the parentheses to the terms inside. This means multiplying the 2 by both x and -3 on the left side of the equation, and multiplying the 3 by both x and -1 on the right side. Let's do it:

$2(x - 3) + 2 = 3(x - 1) + 3$ becomes:

$2x - 6 + 2 = 3x - 3 + 3$

See? Not so bad! We've just expanded the equation to make it easier to work with. Now, let's simplify things a bit further.

Step 2: Combine Like Terms

Now, we're going to combine the like terms on each side of the equation. Like terms are just terms that have the same variable (in this case, x) or are constants (just numbers). On the left side, we have -6 and +2, which are both constants. On the right side, we have -3 and +3, which are also constants. Let's combine them:

$2x - 6 + 2 = 3x - 3 + 3$ becomes:

$2x - 4 = 3x + 0$

Which simplifies to:

$2x - 4 = 3x$

Great! The equation is looking much cleaner now. Next up, we need to get all the x terms on one side of the equation and all the constants on the other side.

Step 3: Isolate the Variable

To isolate the variable x, we need to move all the x terms to one side of the equation and all the constants to the other side. A common way to do this is to subtract 2x from both sides of the equation:

$2x - 4 = 3x$ becomes:

$2x - 4 - 2x = 3x - 2x$

Simplifying, we get:

$-4 = x$

Or, if you prefer:

$x = -4$

Woo-hoo! We've found the value of x that makes the equation true. But wait, there's one more step to make sure we've got it right.

Step 4: Check the Solution

To check our solution, we're going to plug the value we found for x (which is -4) back into the original equation and see if it makes the equation true. Here's the original equation:

$2(x - 3) + 2 = 3(x - 1) + 3$

Now, let's substitute x with -4:

$2((-4) - 3) + 2 = 3((-4) - 1) + 3$

Simplify inside the parentheses:

$2(-7) + 2 = 3(-5) + 3$

Multiply:

$-14 + 2 = -15 + 3$

Add:

$-12 = -12$

It checks out! Both sides of the equation are equal when x = -4. This confirms that our solution is correct.

Step 5: State the Solution Set

Finally, we need to state the solution set. The solution set is simply the set of all values of x that make the equation true. In this case, there's only one value of x that works, which is -4. So, the solution set is:

$\{-4\}$

And that's it! We've successfully found the solution set for the equation $2(x-3) + 2 = 3(x - 1) + 3$. You did it! Great job!

In summary, here are the steps we took:

1. Distribute the numbers outside the parentheses.
2. Combine like terms on each side of the equation.
3. Isolate the variable x.
4. Check the solution by plugging it back into the original equation.
5. State the solution set.

By following these steps, you can solve many similar equations. Keep practicing, and you'll become a pro in no time!

Now, wasn't that fun? Let's summarize the whole process to make sure we've got it down pat. Solving equations like this is a fundamental skill in algebra, so understanding the ins and outs will definitely pay off down the road.

More on Distributing and Combining Terms

Let's really nail down those first couple of steps, because they're super important. When we talk about distributing, we're using the distributive property, which basically says that $a(b + c) = ab + ac$. It's a fancy way of saying multiply the thing outside the parentheses by everything inside. For example, with $2(x - 3)$, we multiply 2 by x to get $2x$, and 2 by -3 to get -6. So, $2(x - 3)$ becomes $2x - 6$.

Combining like terms is all about simplifying each side of the equation. Like terms have the same variable raised to the same power. So, $3x$ and $-5x$ are like terms, but $3x$ and $3x^2$ are not. Constants are also like terms. So, in the expression $2x - 6 + 2$, -6 and 2 are like terms that can be combined to get -4. This gives us $2x - 4$.

Isolating the Variable: Getting X by Itself

Okay, so we've simplified each side of the equation. Now we need to get all the x's on one side and all the numbers on the other. The trick here is to do the same thing to both sides of the equation. This keeps the equation balanced. If we have $2x - 4 = 3x$, we can subtract $2x$ from both sides to get $-4 = x$. See how we did the same operation on both sides?

Another important tip is to remember to do the opposite operation to move terms around. If a term is added on one side, subtract it from both sides. If a term is multiplied on one side, divide both sides by it. Keep the equation balanced, and you'll be golden!

Checking Your Solution: The Sanity Check

Always, always, always check your solution. It's like proofreading an essay. It can save you from silly mistakes. Plug your solution back into the original equation. Simplify each side separately. If both sides are equal, you've got the right answer. If not, go back and check your work.

For example, if we found $x = -4$, we plug that back into $2(x - 3) + 2 = 3(x - 1) + 3$. This gives us $2(-4 - 3) + 2 = 3(-4 - 1) + 3$, which simplifies to $2(-7) + 2 = 3(-5) + 3$, and further simplifies to $-14 + 2 = -15 + 3$, which becomes $-12 = -12$. Since both sides are equal, we know our solution is correct.

Solution Sets: What Are They?

The solution set is just a fancy way of saying the set of all values that make the equation true. Sometimes there's only one solution, like in our example. Sometimes there are no solutions, and sometimes there are infinitely many solutions. When there's only one solution, the solution set is just that one value inside curly braces. So, if $x = -4$ is the only solution, the solution set is $\{-4\}$.

Practice Makes Perfect

The best way to get good at solving equations is to practice. Start with easy problems and gradually work your way up to more difficult ones. The more you practice, the easier it will become. And remember, if you get stuck, don't be afraid to ask for help.

So there you have it, guys! We've walked through how to solve the equation $2(x-3) + 2 = 3(x - 1) + 3$ and how to think about each step along the way. Keep up the great work, and happy problem-solving!