Solving The Inequality: $-9.4 \geq 1.7x + 4.2$

Oct 22, 2025 by Dimemap Team 47 views

$Solving the Inequality: $-9.4 \geq 1.7x + 4.2$$

Hey guys! Today, we're diving into the world of inequalities, specifically how to solve the inequality $-9.4 \geq 1.7x + 4.2$ . Inequalities might seem a bit intimidating at first, but don't worry, we'll break it down step by step. Think of it like solving a regular equation, but with a little twist. Instead of finding a single value for x, we're finding a range of values that make the statement true. So, grab your pencils and let's get started!

Understanding Inequalities

Before we jump into the nitty-gritty, let's quickly recap what inequalities are all about. Inequalities are mathematical statements that compare two expressions using symbols like greater than (>), less than (<), greater than or equal to ( $\geq$ ), and less than or equal to ( $\leq$ ). Unlike equations, which have a single solution (or a few), inequalities have a range of solutions. This range can be represented on a number line or in interval notation.

When we're dealing with inequalities, the goal is the same as with equations: isolate the variable. We want to get x all by itself on one side of the inequality. However, there's one crucial difference: when you multiply or divide both sides of an inequality by a negative number, you need to flip the direction of the inequality sign. Keep that in mind, it's a key step!

Why do we flip the sign? Imagine you have the inequality 2 < 4. That's clearly true. Now, multiply both sides by -1. You get -2 and -4. If you keep the inequality sign the same, you'd have -2 < -4, which is false. To make it true, you need to flip the sign: -2 > -4. This principle applies whenever you multiply or divide by a negative number.

Understanding this concept is super important because it's a common place where people make mistakes when solving inequalities. So, always double-check whether you're multiplying or dividing by a negative, and if you are, remember to flip that sign!

Step-by-Step Solution

Okay, let's tackle our inequality: $-9.4 \geq 1.7x + 4.2$ . We'll go through this methodically, step by step, so you can see exactly how it's done. Don't be afraid to pause and rewind if you need to. The more you practice, the easier this will become.

Step 1: Isolate the Term with x

The first thing we want to do is get the term with x (that's 1.7x in this case) by itself on one side of the inequality. To do that, we need to get rid of the +4.2. How do we do that? We subtract 4.2 from both sides. Remember, whatever we do to one side of the inequality, we have to do to the other to keep things balanced.

So, we have:

$-9.4 - 4.2 \geq 1.7x + 4.2 - 4.2$

Simplifying this gives us:

$-13.6 \geq 1.7x$

Great! We've managed to isolate the term with x. Now, we're one step closer to solving the inequality.

Step 2: Isolate x

Now we need to get x completely alone. It's currently being multiplied by 1.7. To undo that multiplication, we need to divide both sides of the inequality by 1.7.

So, we have:

$\frac{-13.6}{1.7} \geq \frac{1.7x}{1.7}$

Performing the division, we get:

$-8 \geq x$

Or, we can rewrite it as:

$x \leq -8$

There you have it! We've solved the inequality. This tells us that x can be any number less than or equal to -8.

Step 3: Representing the Solution

Now that we've found the solution, it's good practice to represent it in a couple of different ways. This helps us visualize the solution and understand what it means.

Number Line

One way to represent the solution is on a number line. Draw a number line and find -8. Since our solution includes -8 (because of the "less than or equal to" sign), we'll draw a closed circle (or a filled-in dot) at -8. Then, we shade everything to the left of -8, because x can be any number less than -8. The shaded region represents all the possible solutions to our inequality.

Interval Notation

Another way to represent the solution is using interval notation. This is a compact way to write the range of values. Since x is less than or equal to -8, it can go all the way to negative infinity. In interval notation, we write this as:

$(-\infty, -8]$

Notice the parentheses around negative infinity because we can't actually reach infinity. The square bracket around -8 indicates that -8 is included in the solution.

Checking Your Solution

It's always a good idea to check your solution to make sure you haven't made any mistakes. Here's how we can do it for inequalities:

Pick a value within your solution range: Choose any number that's less than or equal to -8. For example, let's pick -10.
Substitute it into the original inequality: Plug -10 in for x in the original inequality:

$-9.4 \geq 1.7(-10) + 4.2$
Simplify:

$-9.4 \geq -17 + 4.2$

$-9.4 \geq -12.8$
Check if the statement is true: Is -9.4 greater than or equal to -12.8? Yes, it is! So, our solution is likely correct.
Pick a value outside your solution range: Choose a number that's greater than -8. Let's pick 0.
Substitute it into the original inequality:

$-9.4 \geq 1.7(0) + 4.2$
Simplify:

$-9.4 \geq 0 + 4.2$

$-9.4 \geq 4.2$
Check if the statement is true: Is -9.4 greater than or equal to 4.2? No, it's not! This confirms that values outside our solution range don't work, which further strengthens our confidence in our solution.

By checking your solution with values inside and outside the range, you can catch any errors you might have made along the way.

Common Mistakes to Avoid

Solving inequalities is pretty straightforward once you get the hang of it, but there are a few common mistakes that people often make. Being aware of these pitfalls can help you avoid them.

Forgetting to Flip the Sign: As we mentioned earlier, this is the big one! Always remember to flip the inequality sign when you multiply or divide both sides by a negative number.
Incorrectly Distributing: If you have an inequality with parentheses, make sure you distribute correctly. For example, if you have 2(x + 3) > 5, you need to multiply both the x and the 3 by 2.
Combining Unlike Terms: Just like with equations, you can only combine like terms. You can't add an x term to a constant term, for example.
Making Arithmetic Errors: Simple arithmetic mistakes can throw off your entire solution. Double-check your calculations, especially when dealing with negative numbers.

By being mindful of these common errors, you can increase your accuracy and solve inequalities with confidence.

Practice Problems

The best way to master solving inequalities is to practice, practice, practice! Here are a few problems for you to try:

$3x + 5 < 14$
$-2x - 7 \geq 3$
$4(x - 1) \leq 8$
$\frac{x}{2} + 1 > 6$

Work through these problems, and don't hesitate to review the steps we covered earlier if you get stuck. Remember to check your solutions, too!

Solving inequalities is a fundamental skill in algebra, and it's something you'll use again and again in more advanced math courses. By understanding the basic principles and practicing regularly, you'll become a pro in no time!

So there you have it, guys! We've successfully solved the inequality $-9.4 \geq 1.7x + 4.2$ . Remember, the key is to isolate the variable while paying close attention to the direction of the inequality sign. Keep practicing, and you'll become an inequality-solving whiz!