Solving Inequalities Finding The Solution Set For 4x - 12 ≤ 16 + 8x
Hey guys! Today, we're diving into a fun little math problem that involves inequalities. Inequalities are just like equations, but instead of an equals sign, we have symbols like "less than or equal to" (≤) or "greater than" (>). So, let's break down this problem step by step and figure out how to solve it together.
Understanding the Inequality
The inequality we're tackling is 4x - 12 ≤ 16 + 8x. Don't let it intimidate you! It looks a bit complex, but we can simplify it. Think of it as a puzzle where we need to find all the values of 'x' that make this statement true. In simpler terms, we need to figure out what numbers we can put in place of 'x' so that when we do the math on both sides, the left side is less than or equal to the right side.
The Goal: Isolate 'x'
Our main goal here is to isolate 'x' on one side of the inequality. This means we want to get 'x' by itself, so we know what its possible values are. We can do this by using some basic algebraic operations – the same ones we use when solving equations. Remember, whatever we do to one side of the inequality, we must do to the other side to keep things balanced. It's like a seesaw; if you add weight to one side, you need to add the same amount to the other to keep it level. Now, let's get into the step-by-step process of isolating 'x'. We'll start by moving the 'x' terms to one side and the constant terms to the other. This will help us simplify the inequality and get closer to our solution. So, stick with me, and let's see how it's done!
Step-by-Step Solution
Let's walk through the steps to solve the inequality 4x - 12 ≤ 16 + 8x. Remember, our goal is to get 'x' by itself on one side of the inequality. Think of it as detective work, where we're gathering clues and piecing them together to solve the mystery of 'x'.
Step 1: Moving 'x' Terms
The first thing we want to do is move all the terms with 'x' to one side of the inequality. To do this, we can subtract 8x from both sides. This keeps the inequality balanced and helps us consolidate the 'x' terms. Subtracting 8x from both sides gives us:
4x - 12 - 8x ≤ 16 + 8x - 8x
Simplifying this, we get:
-4x - 12 ≤ 16
Now, we have all the 'x' terms on the left side, which is a great step forward!
Step 2: Moving Constant Terms
Next, we want to move all the constant terms (the numbers without 'x') to the other side of the inequality. We can do this by adding 12 to both sides. This will cancel out the -12 on the left side and move it to the right side. Adding 12 to both sides gives us:
-4x - 12 + 12 ≤ 16 + 12
Simplifying, we get:
-4x ≤ 28
We're getting closer! Now we have only the 'x' term on the left side and a constant on the right side.
Step 3: Isolating 'x'
Finally, we need to isolate 'x' completely. To do this, we'll divide both sides of the inequality by -4. But here’s a crucial rule to remember: when we divide (or multiply) both sides of an inequality by a negative number, we must flip the direction of the inequality sign. This is super important, so don't forget it! Dividing both sides by -4 gives us:
(-4x) / -4 ≥ 28 / -4
Notice how the "≤" sign has flipped to "≥" because we divided by a negative number. Simplifying, we get:
x ≥ -7
So, what does this mean? It means that 'x' can be any number that is greater than or equal to -7. We've successfully solved the inequality and found the solution set for 'x'. Now, let's think about what specific values of 'x' fit this solution.
Understanding the Solution Set
So, we've found that x ≥ -7. That's fantastic! But what does this actually mean in plain English? It means that any value of 'x' that is greater than or equal to -7 will satisfy the original inequality, which was 4x - 12 ≤ 16 + 8x. Think of it as a club where only certain numbers are allowed to enter – in this case, all the numbers that are -7 or bigger get a free pass.
Visualizing the Solution
Sometimes, it helps to visualize this on a number line. Imagine a number line stretching out from negative infinity on the left to positive infinity on the right. We have a point at -7. Because x can be equal to -7, we fill in a circle at -7 to show that -7 is included in the solution. Then, we draw a line extending to the right from -7, with an arrow indicating that all numbers greater than -7 are also part of the solution. This visual representation makes it super clear which values of 'x' work.
Examples of Values in the Solution Set
Let’s think about some specific numbers. Can 'x' be -7? Yes, it can! Because the inequality includes "equal to," -7 is a valid solution. How about -6? Yep, that works too! Any number greater than -7, like -5, 0, 1, 10, or even 100, will also be a solution. The solution set includes a whole range of numbers. But what about -8? Nope, that doesn't fit because -8 is less than -7. So, it's not part of our club.
Testing the Solution
If you ever want to double-check your answer, you can pick a value from the solution set and plug it back into the original inequality. For example, let's try x = 0. Plugging this into 4x - 12 ≤ 16 + 8x gives us:
4(0) - 12 ≤ 16 + 8(0)
Simplifying, we get:
-12 ≤ 16
This is true! So, 0 is indeed a solution, which confirms our solution set. You can try this with other numbers greater than or equal to -7 to see that they also work. This is a great way to build confidence in your answer and make sure you've got it right. Now that we understand the solution set, let's wrap up and highlight the key takeaways.
Conclusion: The Value of x
Alright, let's wrap things up! We started with the inequality 4x - 12 ≤ 16 + 8x, and through a series of steps, we've determined that the solution is x ≥ -7. This means that any value of 'x' that is greater than or equal to -7 will satisfy the original inequality. This was a fantastic journey through the world of inequalities. We saw how to isolate 'x', how to handle the tricky situation of dividing by a negative number, and how to interpret the solution set.
Key Takeaways
Here are the main things to remember from this problem:
- Isolate 'x': The goal is to get 'x' by itself on one side of the inequality.
- Balance: Whatever operation you do on one side, do it on the other side to keep the inequality balanced.
- Flip the sign: When you multiply or divide by a negative number, flip the direction of the inequality sign.
- Solution set: The solution set includes all values of 'x' that make the inequality true.
- Visualization: Using a number line can help you visualize the solution set.
Importance of Practice
Solving inequalities is a fundamental skill in algebra, and the more you practice, the better you'll get. These skills will come in handy in more advanced math courses and even in real-world situations where you need to make comparisons and decisions based on inequalities. So, keep practicing, keep asking questions, and keep exploring the fascinating world of mathematics! You've got this!
Final Thoughts
We've successfully solved the inequality and found that x ≥ -7. This means that any value of x that is greater than or equal to -7 is a valid solution. Remember the key steps: moving 'x' terms, moving constant terms, and isolating 'x'. And don't forget to flip the inequality sign when dividing or multiplying by a negative number. You guys did great! Keep up the fantastic work, and I'll see you in the next math adventure!
