Graphing The Solution Of (3/4)(x+8) > (1/2)(2x+10) Inequality

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So, you've got this inequality, (3/4)(x+8) > (1/2)(2x+10), and you're tasked with graphing its solution. No sweat, guys! It might look a bit intimidating at first, but we're going to break it down step by step, making it super easy to understand. Think of it as solving a puzzle – each step gets us closer to the final picture, which, in this case, is the graph of the solution on a number line.

Before we even think about graphing, we need to actually solve the inequality. This means isolating 'x' on one side of the inequality sign. We'll use our algebra skills to simplify and rearrange the expression until we get something like 'x > some number' or 'x < some number'. Once we have that, graphing it is a piece of cake. Remember, inequalities are different from equations. An equation has one solution (or a few discrete solutions), while an inequality has a range of solutions. That's why we need a graph – to visualize all the possible values of 'x' that make the inequality true. So, let's dive into the nitty-gritty of solving this inequality and then see how we can represent the solution graphically.

We'll start by tackling the fractions and parentheses. This involves using the distributive property and some basic arithmetic. Don't worry if you feel a bit rusty on these concepts; we'll go through them carefully. The key is to take it one step at a time and make sure you understand each operation before moving on. We'll be multiplying, simplifying, and rearranging terms, all with the goal of getting 'x' all by itself on one side. Once we've done that, we'll have a clear picture of what values 'x' can take. This is where the fun begins – we get to translate our algebraic solution into a visual representation on the number line. Think of the number line as a map, and our solution as the territory we're marking. We'll use open or closed circles and arrows to show exactly which numbers are included in the solution and which ones are not. So, grab your pencils (or your favorite digital drawing tool) and let's get started on this mathematical adventure!

Step 1: Distribute and Simplify

The first thing we need to do is get rid of those parentheses. We do this by distributing the fractions on both sides of the inequality. Remember the distributive property? It basically says that a(b + c) = ab + ac. We're going to use this same principle here. Let's break it down:

On the left side, we have (3/4)(x + 8). We need to multiply (3/4) by both 'x' and 8:

  • (3/4) * x = (3/4)x
  • (3/4) * 8 = 6 (because (3 * 8) / 4 = 24 / 4 = 6)

So, the left side becomes (3/4)x + 6.

Now, let's tackle the right side: (1/2)(2x + 10). Again, we distribute (1/2) to both terms:

  • (1/2) * 2x = x
  • (1/2) * 10 = 5

So, the right side simplifies to x + 5.

Our inequality now looks like this: (3/4)x + 6 > x + 5. See? We've already made some good progress! We've eliminated the parentheses and have a simpler-looking inequality to work with. But we're not done yet. We still need to isolate 'x'. The next step is to get all the 'x' terms on one side and the constant terms on the other.

This step is crucial because it sets the stage for the final isolation of 'x'. By carefully distributing and simplifying, we've transformed the original inequality into a more manageable form. We've essentially cleared the initial hurdles and are now on a clear path toward the solution. Think of it like preparing the ingredients for a recipe – we've got everything chopped and measured, and now we're ready to start cooking (or in this case, solving!). The distributive property is a fundamental tool in algebra, and mastering it is essential for solving more complex equations and inequalities. It's like having a superpower that allows you to break down complicated expressions into smaller, more digestible pieces. So, make sure you're comfortable with this step before moving on. We're building a solid foundation here, and each step is crucial for the overall success of our solution.

Step 2: Isolate the Variable

Now, let's get all the 'x' terms on one side of the inequality and the constants on the other. To do this, we'll use the properties of inequalities, which are very similar to the properties of equality (with one important exception, which we'll discuss later!).

First, let's subtract 'x' from both sides. This will move the 'x' term from the right side to the left side:

(3/4)x + 6 - x > x + 5 - x

Simplifying this gives us:

(3/4)x - x + 6 > 5

Now, we need to combine the 'x' terms. Remember that 'x' is the same as (4/4)x, so we have:

(3/4)x - (4/4)x + 6 > 5

This simplifies to:

(-1/4)x + 6 > 5

Next, let's subtract 6 from both sides to move the constant term to the right side:

(-1/4)x + 6 - 6 > 5 - 6

This simplifies to:

(-1/4)x > -1

We're getting closer! We now have the 'x' term isolated on the left side, but it's still being multiplied by a fraction. The final step in isolating 'x' is to get rid of that fraction.

This step is where the one important difference between inequalities and equations comes into play. When we multiply or divide both sides of an inequality by a negative number, we need to flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the numbers on the number line. For example, 2 < 3, but -2 > -3.

So, to get rid of the (-1/4) multiplying 'x', we need to multiply both sides by -4. And because we're multiplying by a negative number, we need to flip the inequality sign:

(-4) * (-1/4)x < (-4) * (-1)

This simplifies to:

x < 4

Woohoo! We've done it! We've solved the inequality. Our solution is x < 4. This means that any number less than 4 will make the original inequality true. But we're not quite finished yet. We still need to graph this solution on a number line.

This process of isolating the variable is like peeling back the layers of an onion – each step reveals a little more of the core. We've used a combination of arithmetic and algebraic manipulations to get to this point, and it's a testament to the power of these tools. The rule about flipping the inequality sign when multiplying or dividing by a negative number is a crucial detail to remember. It's a subtle but important difference between working with equations and inequalities. So, make sure you've got that firmly in your mind. Now that we have our solution, x < 4, we're ready to visualize it on the number line. This is where the abstract algebra transforms into a concrete graphical representation.

Step 3: Graph the Solution

Okay, we've got our solution: x < 4. This means we need to represent all the numbers that are less than 4 on a number line. Here's how we do it:

  1. Draw a Number Line: Start by drawing a straight line. Mark zero in the middle, and then mark some numbers to the left and right of zero, like -3, -2, -1, 1, 2, 3, 4, 5. Make sure the numbers are evenly spaced.
  2. Locate the Critical Value: Our critical value is 4, because that's the number our solution is based on. Find 4 on your number line.
  3. Open or Closed Circle?: This is important! Because our inequality is x < 4 (less than, not less than or equal to), we use an open circle at 4. An open circle means that 4 itself is not included in the solution. If our inequality were x ≤ 4 (less than or equal to), we would use a closed circle to indicate that 4 is included.
  4. Shade the Correct Direction: We want to represent all the numbers less than 4. On a number line, numbers get smaller as we move to the left. So, we'll shade the number line to the left of the open circle at 4. This shaded region represents all the values of 'x' that satisfy the inequality.
  5. Draw an Arrow: To show that the shading continues infinitely in the direction of the solution, we draw an arrow at the end of the shaded region. In this case, we draw an arrow pointing to the left, indicating that all numbers less than 4 are solutions.

That's it! You've graphed the solution to the inequality. The open circle at 4 and the shaded line with an arrow pointing to the left visually represent all the values of 'x' that make the inequality (3/4)(x+8) > (1/2)(2x+10) true.

Graphing the solution is like painting a picture of the answer. It takes the abstract algebraic solution and makes it visually accessible. The open and closed circles are like signposts, telling us whether the endpoint is included or excluded. The shading is the territory of the solution, showing us all the possible values that satisfy the inequality. This visual representation is incredibly powerful because it allows us to quickly grasp the range of solutions and understand the concept of inequalities in a more intuitive way. So, practice graphing inequalities, guys! It's a fundamental skill in mathematics, and it opens the door to understanding more advanced concepts.

Conclusion

So, we've successfully navigated the world of inequalities! We started with a somewhat complex-looking inequality, (3/4)(x+8) > (1/2)(2x+10), and we've taken it all the way to a beautiful graphical representation of its solution. We've learned how to:

  • Distribute and simplify expressions.
  • Isolate the variable using the properties of inequalities (remembering to flip the sign when multiplying or dividing by a negative number!).
  • Graph the solution on a number line using open and closed circles and shading to indicate the range of values that satisfy the inequality.

This process is a fundamental skill in algebra, and it's applicable to a wide range of mathematical problems. Inequalities are used to represent situations where things are not necessarily equal, which is very common in the real world. Think about things like speed limits, budget constraints, or the temperature range for a comfortable day – these are all situations that can be modeled using inequalities.

By mastering the techniques we've covered in this guide, you've equipped yourself with a powerful tool for solving these kinds of problems. And more importantly, you've developed a deeper understanding of how mathematical concepts can be used to represent and solve real-world situations. So, keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is vast and fascinating, and you've just taken another step on your journey of discovery. Remember, practice makes perfect, guys! The more you work with inequalities, the more comfortable and confident you'll become. And don't be afraid to ask questions and seek help when you need it. We're all in this together, learning and growing, one inequality at a time. So go forth and conquer those number lines! You've got this!