Solving 1.5x + 3.75 ≥ 5.5 A Step By Step Guide

Hey guys! Inequalities might seem like tricky puzzles at first, but trust me, once you get the hang of them, they're super manageable. This guide is all about breaking down the inequality 1.5x + 3.75 ≥ 5.5 and finding out which answer choice—whether it's -1, 1, 1.5, or 0—actually works as a solution. We're going to take a step-by-step approach, so you'll not only get the answer but also understand the why behind it. Think of this as your friendly guide to conquering inequalities!

Understanding Inequalities: The Basics

Before we jump into solving our specific inequality, let’s make sure we’re all on the same page with the basics. What exactly is an inequality? Well, unlike an equation that uses an equals sign (=) to show that two expressions are exactly the same, an inequality uses symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to) to show a relationship where two expressions might not be exactly equal.

So, when we see something like 1.5x + 3.75 ≥ 5.5, it means we're looking for values of x that make the left side of the expression either greater than or equal to the right side. It's like we're trying to find a range of numbers that fit the condition, rather than just one specific number. This is a crucial concept to grasp because it's the foundation for everything else we'll be doing.

Think of it like this: If you have a minimum height requirement for a rollercoaster, that's an inequality in action! You need to be at least a certain height to ride, meaning your height has to be greater than or equal to the minimum. That's the same idea we're working with here, just with numbers and algebraic expressions. Understanding this basic principle makes the whole process a lot less intimidating. You've got this!

Key Concepts and Symbols

Let's dive a bit deeper into the key concepts and symbols involved in inequalities. It's super important to be crystal clear on what each symbol means, as this will guide how we solve and interpret the solutions. Remember those symbols we talked about earlier? Let's break them down:

• > means "greater than." For example, 5 > 3 (5 is greater than 3).
• < means "less than." For example, 2 < 7 (2 is less than 7).
• ≥ means "greater than or equal to." This is the symbol we see in our problem, 1.5x + 3.75 ≥ 5.5. It means the left side can be bigger than 5.5, but it can also be exactly 5.5. Think of it as a range that includes the endpoint.
• ≤ means "less than or equal to." Similar to the above, this means the left side can be smaller than the right side, or it can be equal to it.

Now, let's talk about the concept of a solution. A solution to an inequality is any value that, when plugged in for the variable (in our case, x), makes the inequality true. Unlike an equation where we often find one specific solution, inequalities usually have a range of solutions. This is because there are multiple numbers that can satisfy the condition of being greater than, less than, greater than or equal to, or less than or equal to a certain value.

Understanding these symbols and the idea of a solution set is like having the right tools for the job. You can't build a house without knowing what a hammer and a nail are for, right? Same goes for inequalities! So, with these concepts in your toolkit, you're well-prepared to tackle the problem at hand.

Solving the Inequality 1.5x + 3.75 ≥ 5.5

Okay, guys, let's roll up our sleeves and get into the nitty-gritty of solving this inequality! We're going to tackle 1.5x + 3.75 ≥ 5.5 using the same principles we'd use to solve a regular equation, but with a tiny twist or two. Don't worry, it's nothing you can't handle.

The main goal here is to isolate x on one side of the inequality. Think of it like peeling away the layers of an onion until we get to the core. We want to get x all by itself so we can see what values make the inequality true. Here’s how we’ll do it, step-by-step:

Step 1: Isolate the Term with x

Our first move is to get the term with x (that's 1.5x) all alone on one side. To do this, we need to get rid of the + 3.75 that's hanging out with it. How do we do that? We use the inverse operation! Since we're adding 3.75, we'll subtract 3.75 from both sides of the inequality. Remember, what we do to one side, we gotta do to the other to keep things balanced.

So, our inequality becomes:

1.5x + 3.75 - 3.75 ≥ 5.5 - 3.75

This simplifies to:

1.5x ≥ 1.75

See? We're already one step closer! The term with x is now more isolated, and we're making great progress.

Step 2: Solve for x

Now that we've got 1.5x ≥ 1.75, we need to get x completely by itself. Right now, x is being multiplied by 1.5. So, what's the inverse operation we need to use? You guessed it—division! We're going to divide both sides of the inequality by 1.5 to undo that multiplication.

Here’s what it looks like:

(1.5x) / 1.5 ≥ 1.75 / 1.5

When we do the math, we get:

x ≥ 1.1666...

Wait a minute… what's with that repeating decimal? It might be a bit tricky to work with in its current form. Let's simplify it. 1.75 divided by 1.5 is actually equal to 7/6. So, we can rewrite our inequality as:

x ≥ 7/6

Or, if we want to express it as a decimal, we can round it to two decimal places for simplicity:

x ≥ 1.17 (approximately)

So, what does this mean? It means that any value of x that is greater than or equal to 1.17 (or 7/6) will make our original inequality true. We've successfully solved for x!

Important Note: Flipping the Inequality Sign

Now, here's a little heads-up that's super important to remember when dealing with inequalities: If you ever multiply or divide both sides of an inequality by a negative number, you have to flip the direction of the inequality sign. This is a crucial rule because multiplying or dividing by a negative number changes the order of the numbers on the number line.

For example, if you have -2x > 4, and you divide both sides by -2, you need to flip the > sign to <, resulting in x < -2.

Luckily, in our specific problem, we didn't have to worry about this because we only divided by a positive number (1.5). But it's always good to keep this rule in the back of your mind for future inequality adventures!

Checking the Answer Choices

Alright, we've solved the inequality and found that x ≥ 1.17 (approximately). Now, the fun part: let's see which of the answer choices—-1, 1, 1.5, and 0—actually fit the bill. We're going to plug each one into the original inequality, 1.5x + 3.75 ≥ 5.5, and see if it holds true.

This is a super important step because it's like double-checking your work. You wouldn't submit a puzzle without making sure all the pieces fit, right? The same goes for inequalities. Let's get to it!

Testing Each Option

Let's systematically test each answer choice. This is where we put our solution to the test and see which value(s) satisfy the original inequality.

1. x = -1

   Substitute -1 into the inequality:

   1. 5(-1) + 3.75 ≥ 5.5 -1. 5 + 3.75 ≥ 5.5
   2. 25 ≥ 5.5

   This is not true. So, -1 is not a solution.

2. x = 1

   Substitute 1 into the inequality:

   1. 5(1) + 3.75 ≥ 5.5
   2. 5 + 3.75 ≥ 5.5
   3. 25 ≥ 5.5

   This is not true. So, 1 is not a solution either.

3. x = 1.5

   Let's plug in 1.5:

   1. 5(1.5) + 3.75 ≥ 5.5
   2. 25 + 3.75 ≥ 5.5 6 0 ≥ 5.5

   This is true! So, 1.5 is a solution to the inequality.

4. x = 0

   Finally, let's test 0:

   1. 5(0) + 3.75 ≥ 5.5
   2. 75 ≥ 5.5

   This is not true. Therefore, 0 is not a solution.

The Verdict

After carefully testing each option, we've found that only one answer choice makes the inequality true: x = 1.5. This aligns perfectly with our solution x ≥ 1.17 (approximately), as 1.5 is indeed greater than 1.17. So, we've not only solved the inequality but also confirmed our solution by checking the given options. High five!

Conclusion: Mastering Inequalities

Woohoo! You guys have made it to the end, and you've officially conquered this inequality problem. We took on the challenge of 1.5x + 3.75 ≥ 5.5, and we nailed it by systematically solving for x and checking each answer choice. You've learned how to isolate variables, handle those tricky inequality symbols, and even remember the golden rule about flipping the sign when multiplying or dividing by a negative number. That's a lot of math power in your hands!

But more than just getting the right answer, you've gained a deeper understanding of why the solution works. You know that an inequality represents a range of possible values, and you've seen firsthand how to find those values. You've practiced the art of substituting and verifying, which is a crucial skill in all areas of math.

So, what's the takeaway here? Inequalities might seem daunting at first, but with a step-by-step approach and a solid grasp of the fundamentals, they become totally manageable. Keep practicing, keep exploring, and most importantly, keep believing in your math abilities. You've got this! And who knows? Maybe the next inequality you solve will unlock an even bigger mathematical adventure. 😉