Understanding Inequalities: Finding Solutions for 5x + 7 < 20

Cracking the Code: Solving Inequalities with Ease

Hey there! If you're one of those folks who dread math but know it’s a necessary evil for college credits, you're not alone. Inequalities might seem like a foreign language, but trust me, they’re not as scary as they sound. Plus, they’re pretty useful—much like knowing how to change a tire or being able to figure out the tip at a restaurant without pulling out your phone. Let’s tackle an inequality together using a simple equation, and I promise it’ll start to feel a lot less intimidating.

What’s the Problem?

Let’s break it down: we’re looking at an inequality expressed like this—5x + 7 < 20. Your first thought might be, "What in the world does this mean?" Essentially, this is saying that what’s on the left side of the equation (5x + 7) needs to be less than 20. The goal here is to isolate the variable (x). Why? Because we're looking to find all of the possible values (x) can take that make the statement true.

Now, you might be wondering, "How do I even begin?" Well, let’s roll up our sleeves and get to work!

Step One: Rearranging the Furniture

First thing’s first, let’s move that pesky 7 to the other side of the inequality. To do this, we subtract 7 from both sides. It’s all about balance, right? So here’s what we get:

[ 5x < 20 - 7 ]

This simplifies to:

[ 5x < 13 ]

A Little Side Note

Before I move on, let’s pause and think about what's happening here. When you’re isolating a variable in inequalities, you’re performing the same operations throughout—yes, just like you would with equations. This is essential for maintaining the truth of the statement.

Step Two: Dividing for Clarity

Next up, we’ll divide both sides by 5 to solve for (x):

[ x < \frac{13}{5} ]

Now, plugging that into a calculator (or just doing a little mental math if you're feeling brave), we find this is equal to:

[ x < 2.6 ]

Here comes the fun part—what’s 2.6 in terms of whole numbers? We typically round down when it comes to these types of inequalities. So instead of saying (x < 2.6), we say (x < 2)—and there you have it! But let’s take a moment to check our work.

Putting it to the Test

Got a second? Here's where we make sure our answer makes sense. Let’s use some of those choices provided:

• A. (x < 2)

• B. (x < 3)

• C. (x < 4)

• D. (x < 5)

If we use (x = 2), we get:

[ 5(2) + 7 = 10 + 7 = 17 ]

And sure enough, 17 is less than 20!

Let’s try (3).

[ 5(3) + 7 = 15 + 7 = 22 ]

Boom—22 is not less than 20!

And with (4):

[ 5(4) + 7 = 20 + 7 = 27 ]

Not less than 20 either!

The only option that worked was A: (x < 2). So, overall, that's the answer we would go with, right?

A Quick Reflection

Numbers can sometimes feel like they’re out to trick you, but here's a little secret: once you understand the steps to isolate your variable, solving inequalities gets a lot simpler. Less confusion, more clarity!

Why Does This Matter?

So why should you care about inequalities like (5x + 7 < 20)? Well, think of inequalities as the foundation for so many concepts—like those in statistics and calculus—that help make sense of our everyday world!

Just imagine you’re planning a party. You might have a limit on how many guests you can invite—this is similar to an inequality! Or, if you're budgeting for that same shindig, keeping your total costs under a certain amount could be seen through an inequality lens too. There’s practicality in these problems; they’re not just about solving for (x).

Wrapping it Up

So, there you have it—a simple guide through the world of inequalities. You figured out that to solve (5x + 7 < 20), we can reduce it down to (x < 2).

Every time you break down a math problem into simpler chunks, you're building skills that stretch far beyond the classroom. You're sharpening your analytical skills, problem-solving prowess, and maybe even a little sense of accomplishment. And let's admit it, feeling confident in math can do wonders for your overall college experience.

Next time an inequality rolls your way, don’t panic. Just remember the steps: isolate that variable, test your solutions, and keep an eye on the real-life applications—it’ll make the journey that much more meaningful. Happy solving!