Solving Inequalities -6x + 3 > -2x A Step-by-Step Guide

Hey guys! Today, we're diving into a common type of math problem: solving linear inequalities. Specifically, we're going to break down the inequality $-6x + 3 > -2x$ step-by-step, so you can tackle similar problems with confidence. We'll not only find the solution but also understand the reasoning behind each step. So, grab your pencils and let's get started!

Understanding Linear Inequalities

Before we jump into the solution, let's make sure we're all on the same page about linear inequalities. Unlike linear equations, which have a single solution, linear inequalities represent a range of values. They involve comparison symbols like '>', '<', '≥', and '≤'. Solving an inequality means finding all the values of the variable that make the inequality true. Think of it like finding all the numbers that fit a certain condition, rather than just one specific number.

The inequality $-6x + 3 > -2x$ is a linear inequality because the variable 'x' is raised to the power of 1. The goal here is to isolate 'x' on one side of the inequality, just like we do with equations, but with a small twist – we need to be mindful of what happens when we multiply or divide by a negative number. This is a crucial point that we'll highlight later.

Why are inequalities important? You might be wondering, why bother with inequalities? Well, they're incredibly useful in real-world scenarios. Imagine you're trying to figure out how many hours you need to work to earn a certain amount of money, or you're trying to stay within a budget. Inequalities help you model these situations where you're looking for a range of possibilities rather than a single answer. They are also fundamental in fields like optimization, where you're trying to find the best possible outcome within certain constraints.

Step-by-Step Solution

Now, let's get down to business and solve the inequality $-6x + 3 > -2x$. We'll break it down into manageable steps, so it's easy to follow along. Remember, the key is to isolate 'x' on one side.

Step 1: Combining 'x' Terms

Our first goal is to get all the terms with 'x' on the same side of the inequality. To do this, we can add $6x$ to both sides of the inequality. This is similar to what we do with equations – we perform the same operation on both sides to maintain the balance. By adding $6x$ to both sides, we eliminate the $-6x$ term on the left side:

$-6x + 3 > -2x$

Add $6x$ to both sides:

$-6x + 3 + 6x > -2x + 6x$

This simplifies to:

$3 > 4x$

So far, so good! We've managed to get all the 'x' terms on the right side, which is a good step towards isolating 'x'.

Step 2: Isolating 'x'

Next, we need to isolate 'x' completely. Currently, 'x' is being multiplied by 4. To undo this multiplication, we'll divide both sides of the inequality by 4. Again, this is a standard algebraic manipulation that keeps the inequality balanced:

$3 > 4x$

Divide both sides by 4:

rac{3}{4} > rac{4x}{4}

This simplifies to:

rac{3}{4} > x

We're almost there! We've successfully isolated 'x', but the inequality is written with 'x' on the right side. While this is technically correct, it's often easier to understand the solution if 'x' is on the left side. So, let's rewrite the inequality.

Step 3: Rewriting the Inequality

To rewrite the inequality with 'x' on the left side, we simply flip the entire inequality. This means that rac{3}{4} > x is the same as x < rac{3}{4}. It's crucial to remember that when you flip the sides of an inequality, you also need to flip the inequality sign. This is a fundamental rule that ensures the inequality remains true.

So, we have:

x < rac{3}{4}

This is our final solution! It tells us that any value of 'x' that is less than rac{3}{4} will make the original inequality true. We've successfully solved the inequality.

Common Pitfalls and Key Considerations

Before we celebrate our success, let's talk about some common mistakes people make when solving inequalities and some key things to keep in mind. Avoiding these pitfalls will help you solve inequalities accurately and efficiently.

Pitfall 1: Forgetting to Flip the Sign

The most common mistake when solving inequalities is forgetting to flip the inequality sign when multiplying or dividing by a negative number. This is a crucial rule that must be followed. For example, if you have $-2x > 4$, you need to divide both sides by -2, and you must flip the inequality sign, resulting in $x < -2$. Failing to do so will lead to an incorrect solution. Always double-check your work, especially when dealing with negative numbers.

Pitfall 2: Misunderstanding the Solution Set

Another common issue is not fully understanding what the solution represents. An inequality like x < rac{3}{4} means that any number less than rac{3}{4} is a solution. This is an infinite set of numbers! It's important to visualize this solution on a number line to get a better grasp of it. On a number line, you would represent this solution with an open circle at rac{3}{4} and an arrow extending to the left, indicating all values less than rac{3}{4}.

Pitfall 3: Incorrectly Combining Like Terms

Just like with equations, correctly combining like terms is essential for solving inequalities. Make sure you're only combining terms that have the same variable and exponent. For instance, you can combine $-6x$ and $-2x$, but you can't combine $-6x$ and 3. A simple mistake in combining like terms can throw off the entire solution, so pay close attention to the details.

Key Consideration: Checking Your Solution

A great way to ensure your solution is correct is to check it by plugging a value from your solution set back into the original inequality. For example, since our solution is x < rac{3}{4}, we can choose a value less than rac{3}{4}, like 0, and plug it into the original inequality:

$-6(0) + 3 > -2(0)$

$3 > 0$

This is true, which gives us confidence that our solution is correct. If we had chosen a value that didn't satisfy the original inequality, we would know that we made a mistake somewhere and need to re-examine our steps. Checking your solution is a simple yet powerful way to avoid errors.

Conclusion: Mastering Inequalities

So, we've successfully solved the inequality $-6x + 3 > -2x$ and found the solution to be x < rac{3}{4}. We've also discussed the importance of understanding inequalities, the step-by-step process of solving them, and common pitfalls to avoid. Remember, practice makes perfect! The more you work with inequalities, the more comfortable and confident you'll become.

The key takeaways are: isolate the variable, remember to flip the sign when multiplying or dividing by a negative number, understand the solution set, and always check your answer. With these principles in mind, you'll be well-equipped to tackle any linear inequality that comes your way. Keep practicing, and you'll become a master of inequalities in no time! Good luck, guys! And remember, math can be fun when you break it down step by step.