Solve $2(4+2x) \geq 5x+5$ Inequality

Hey guys! Today, we're diving deep into the world of inequalities with a specific problem: solve the inequality $2(4+2x) \geq 5x+5$. Inequalities are super important in math because they help us understand ranges of values rather than just single points. Think of it like this: instead of saying 'x is exactly 3', an inequality might say 'x is greater than or equal to 3', which opens up a whole universe of possibilities for x. This particular inequality involves a bit of distribution and then some basic algebraic manipulation to isolate our variable, 'x'. We're going to break it down step-by-step, making sure we cover all the bases so you can tackle similar problems with confidence. So, grab your notebooks, maybe a snack, and let's get this math party started! We'll be focusing on getting 'x' all by itself on one side of the inequality sign, and remember, the golden rule with inequalities is to be super careful when multiplying or dividing by a negative number – it flips the inequality sign on you! Don't worry, we'll highlight that as we go. Our main goal here is to find the set of all 'x' values that make the statement $2(4+2x) \geq 5x+5$ true. This isn't just about finding a single number; it's about defining a range. So, let's roll up our sleeves and get this done.

Step 1: Distribute and Simplify Both Sides

Alright, the first thing we gotta do when we look at the inequality $2(4+2x) \geq 5x+5$ is to simplify it as much as possible. That '2' chilling outside the parentheses on the left side? It needs to multiply both terms inside. This is the distributive property, a fundamental tool in our algebraic toolbox. So, $2 \times 4$ gives us $8$, and $2 \times 2x$ gives us $4x$. Now, the left side of our inequality transforms from $2(4+2x)$ into $8 + 4x$. The right side, $5x+5$, is already as simple as it gets. So, our inequality now looks like this: $8 + 4x \geq 5x+5$. See? We've already made progress! The expression is cleaner, and it's easier to see the terms involving 'x' and the constant terms. This simplification step is crucial because it removes any ambiguity and sets us up for the next phase: gathering our 'x' terms and our constant terms. Always aim to simplify first, guys. It's like preparing your ingredients before you start cooking – makes the whole process smoother. We're aiming to get all the 'x' terms on one side and all the numbers on the other. This is a common strategy for solving equations and inequalities, and it works like a charm. So, keep this simplified form in mind: $8 + 4x \geq 5x+5$. We're well on our way to figuring out what values of 'x' satisfy this condition.

Step 2: Isolate the Variable 'x'

Now that we've got our simplified inequality, $8 + 4x \geq 5x+5$, it's time to get that pesky 'x' all by itself. The general strategy here is to move all the terms containing 'x' to one side of the inequality and all the constant terms (the numbers without any 'x') to the other side. You can choose which side to move them to, but sometimes one way is slightly easier or leads to fewer negative signs, which can reduce the chance of errors. Let's decide to move the 'x' terms to the right side and the constant terms to the left side. To move the $4x$ from the left side, we subtract $4x$ from both sides of the inequality. Remember, whatever you do to one side, you must do to the other to maintain the balance. So, we have:

$(8 + 4x) - 4x \geq (5x + 5) - 4x$

This simplifies to:

$8 \geq x + 5$

Great! Now all our 'x' terms are on the right. The next step is to get the constant terms together. We need to move the '+5' from the right side over to the left. To do this, we subtract $5$ from both sides of the inequality:

$8 - 5 \geq (x + 5) - 5$

This leaves us with:

$3 \geq x$

Boom! We've successfully isolated 'x'. We now have the inequality in its simplest form: $3 \geq x$. This means that 'x' must be less than or equal to $3$. We've effectively solved the inequality! It's pretty satisfying when you see the variable all alone, right? This final form tells us the entire range of numbers that will make the original statement true. We're almost done, but it's always good practice to check our answer, especially with inequalities. So, stick around for the next step where we'll do just that.

Step 3: Checking Your Solution

So, we've solved the inequality $2(4+2x) \geq 5x+5$ and arrived at the solution $3 \geq x$, which is the same as saying $x \leq 3$. Now, to be absolutely sure we haven't messed anything up – and trust me, it happens to the best of us – we should check our answer. This involves picking a few test values for 'x' that fit our solution and one value that doesn't fit, and plugging them back into the original inequality: $2(4+2x) \geq 5x+5$.

Case 1: A value that should work (x = 3).

Our solution says 'x' can be equal to 3. Let's substitute $x=3$ into the original inequality:

$2(4 + 2*3) \geq 5*3 + 5$

$2(4 + 6) \geq 15 + 5$

$2(10) \geq 20$

$20 \geq 20$

This statement is true! Since $20$ is indeed greater than or equal to $20$, our solution holds for $x=3$. This is a great sign!

Case 2: A value that should work (x = 0).

Let's pick another value less than 3, say $x=0$. This should also satisfy the inequality.

$2(4 + 2*0) \geq 5*0 + 5$

$2(4 + 0) \geq 0 + 5$

$2(4) \geq 5$

$8 \geq 5$

This statement is also true! $8$ is greater than $5$. Excellent!

Case 3: A value that should NOT work (x = 4).

Now, let's pick a value that is greater than 3, say $x=4$. According to our solution $x \leq 3$, this value should not make the inequality true.

$2(4 + 2*4) \geq 5*4 + 5$

$2(4 + 8) \geq 20 + 5$

$2(12) \geq 25$

$24 \geq 25$

This statement is false! $24$ is not greater than or equal to $25$. This confirms that our solution range is correct.

Conclusion of the check: By testing values that satisfy our solution ($x=3$ and $x=0$) and one value that does not ($x=4$), we've confirmed that our solution $x \leq 3$ is accurate. This checking process is a lifesaver for avoiding silly mistakes and building confidence in your answers. Always take a moment to verify your work, especially when dealing with inequalities!

Final Answer and Interpretation

After carefully working through the steps, we have determined that the solution to the inequality $2(4+2x) \geq 5x+5$ is $x \leq 3$. What does this mean in plain English, guys? It means that any number you choose for 'x' that is less than or equal to $3$ will make the original statement true. For instance, if you plug in $x=2$, $x=-10$, or even $x=3$ itself, the inequality will hold. However, if you pick any number greater than $3$, like $x=3.1$ or $x=100$, the inequality will be false. This range, $x \leq 3$, represents an infinite set of numbers on the number line, starting at $3$ and extending infinitely to the left. This is a core concept in understanding algebraic inequalities – they define regions or sets of values, not just single points. You might also see this represented in interval notation. Since 'x' can be any real number less than or equal to 3, the interval is $(-\infty, 3]$. The parenthesis indicates that negative infinity is not included (as it's not a number), and the square bracket indicates that $3$ is included in the solution set. So, whether you write it as $x \leq 3$, or $(-\infty, 3]$, you're describing the same set of values. You've successfully solved the inequality, simplified it, isolated the variable, and even checked your work. That's a solid mathematical victory! Keep practicing these types of problems, and you'll become a pro in no time. Remember the key steps: simplify, isolate, and check. You got this!