Solving Inequalities: A Step-by-Step Guide

Dec 5, 2025 by Admin 43 views

Hey math enthusiasts! Today, we're diving into the world of inequalities. Specifically, we'll solve the inequality $-1+6(-1-3x) > -39-2x$ . This might seem a bit daunting at first, but trust me, it's all about following the right steps. By the end of this, you'll be a pro at solving these types of problems. So, let's get started, shall we? This topic is super important because inequalities are used everywhere, from calculating your budget to understanding how fast something is moving. If you're struggling with algebra or just want to brush up on your skills, this guide is perfect for you. We'll break down each part and show you how to find the answer. Don't worry if it's new to you – we'll go slowly and make sure you understand every step of the way. Let’s get our hands dirty and figure out how to solve this inequality. We'll start by looking at what an inequality really is and then slowly figure out how to solve it. It's like a puzzle, and you get to be the detective! So, get ready to grab your pencils and get ready to crack this code. Ready to start? Let's go!

Understanding the Basics of Inequalities

Before we jump into solving the specific inequality, let's quickly recap what an inequality is all about. Basically, inequalities are mathematical statements that compare two expressions using symbols like:

> (greater than)
< (less than)
≥ (greater than or equal to)
≤ (less than or equal to)
≠ (not equal to)

Unlike equations, which use an equals sign (=), inequalities show a range of possible values. The solution to an inequality is not just one number but a set of numbers that satisfy the inequality. For instance, if you have an inequality like x > 2, it means x can be any number greater than 2 (2.0001, 3, 100, etc.). Think of it as opening up a range of possibilities, not just one specific point. This is super useful in real-life scenarios, like when you’re budgeting. Maybe you need to spend less than a certain amount of money or you need to make more than a certain amount of money to save. It also pops up in science and engineering when looking at speed or measuring temperature. Recognizing these symbols is the first step towards understanding how to solve inequalities. So, remember them well, they are your best friends in the mathematical journey.

Inequality Symbols and Their Meanings

Let’s make sure we're on the same page with the inequality symbols. It’s like learning the language before you can read the book. Each symbol gives us different information about the relationship between two values.

Greater than (>): The expression on the left side is larger than the expression on the right side.
Less than (<): The expression on the left side is smaller than the expression on the right side.
Greater than or equal to (≥): The expression on the left side is larger than, or equal to, the expression on the right side. This means the solution can include the value.
Less than or equal to (≤): The expression on the left side is smaller than, or equal to, the expression on the right side. Like the greater than or equal to, the solution can include the value.
Not equal to (≠): The expression on the left side is not the same as the expression on the right side.

Knowing what each of these symbols means is super important for understanding and solving inequalities. Without this, you might get confused, which is normal. But with a little practice, it'll become second nature! Always check what each of these means because they determine the values that satisfy an inequality.

Step-by-Step Solution to the Inequality

Alright, let’s get down to the nitty-gritty of solving the inequality $-1 + 6(-1 - 3x) > -39 - 2x$ . We’ll break this down into several steps to make it as easy as pie. Each step builds on the previous one, so try not to skip ahead. Take your time, and you'll do great! We are going to go through these steps:

Simplify the Left Side: We’ll start by simplifying the left side of the inequality. This involves distributing the 6 across the terms inside the parentheses.
Combine Like Terms: After distributing, we'll combine any like terms on each side of the inequality.
Isolate the Variable: Next, we need to get all the x terms on one side of the inequality and the constant terms on the other side.
Solve for x: Finally, we'll isolate x by dividing both sides of the inequality by the appropriate number.

Ready? Let’s start with the first step!

Step 1: Simplify the Left Side

First, we need to deal with those parentheses. Remember the order of operations (PEMDAS/BODMAS)? We need to multiply before we can add or subtract. So, let’s distribute the 6 across the terms inside the parentheses:

$6 * -1 = -6$
$6 * -3x = -18x$

So the left side of the inequality $-1 + 6(-1 - 3x)$ becomes $-1 - 6 - 18x$ . Now, we can combine the constants: $-1 - 6 = -7$ .

The left side of the inequality now simplifies to $-7 - 18x$ . So our inequality looks like this:

$-7 - 18x > -39 - 2x$

Great job! You’ve successfully simplified the left side of the inequality.

Step 2: Combine Like Terms

In our simplified inequality $-7 - 18x > -39 - 2x$ , we can see that we have a few constants and x terms. Now we'll combine like terms if possible, which in this case, we'll only need to focus on moving the x terms and constants.

Now we want to start getting the x terms on one side and the constant terms on the other. This usually makes it easier to solve.

Step 3: Isolate the Variable

Our next goal is to get all the x terms on one side and the constant terms on the other. Let’s add $2x$ to both sides of the inequality to get the x terms together:

$-7 - 18x + 2x > -39 - 2x + 2x$

This simplifies to:

$-7 - 16x > -39$

Now, let's get the constant terms on the other side. Add 7 to both sides:

$-7 - 16x + 7 > -39 + 7$

Which simplifies to:

$-16x > -32$

Almost there!

Step 4: Solve for x

Now, we need to isolate $x$ . We do this by dividing both sides of the inequality by -16. Remember that when you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign.

So, we divide both sides by -16 and flip the inequality sign:

$rac{-16x}{-16} < rac{-32}{-16}$

This simplifies to:

$x < 2$

And there we have it! The solution to our inequality is $x < 2$ .

Checking Your Work and Understanding the Solution

Now that we've found our answer, $x < 2$ , let's make sure it makes sense. This part is crucial, as it helps you confirm that your solution is correct.

Checking the Solution

One way to check your solution is to pick a number that satisfies the inequality $x < 2$ and plug it back into the original inequality. Let's pick x = 1. Remember, 1 is less than 2, so it satisfies our solution. Now, let’s substitute x = 1 into the original inequality:

$-1 + 6(-1 - 3(1)) > -39 - 2(1)$

Simplify the left side:

$-1 + 6(-1 - 3) > -39 - 2$

$-1 + 6(-4) > -41$

$-1 - 24 > -41$

$-25 > -41$

This is true! So, our solution is likely correct. This shows that when x = 1, the inequality holds true.

Understanding the Solution Set

The solution $x < 2$ means that any number less than 2 will satisfy the original inequality. This is an infinite set of numbers. On a number line, this would be represented by an open circle at 2 (since 2 is not included) and an arrow pointing to the left, indicating all numbers less than 2.

Conclusion: Mastering Inequalities

Awesome work, everyone! You've successfully solved the inequality $-1 + 6(-1 - 3x) > -39 - 2x$ , and the solution is $x < 2$ . We covered the basics of inequalities, the steps to solve them, and how to check your work. Remember, practice is key. The more you solve these problems, the easier they will become. Keep up the great work, and you'll be acing these problems in no time! Remember to always check your answers to make sure they're correct. Keep practicing, and you'll be a pro in no time. If you have any more questions, feel free to ask. Keep up the great work!