Solving The Inequality: $-2(x+3) > X+6$ For $x$

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Hey guys! Let's dive into solving this inequality problem together. Inequalities might seem tricky at first, but once you break them down step-by-step, they become much easier to handle. In this article, we'll walk through the process of solving the inequality $-2(x+3) > x+6$ for $x$. We'll cover each step in detail, ensuring you understand the logic and math behind it. So, grab your pencils and notebooks, and let's get started!

Understanding Inequalities

Before we jump into solving this specific problem, let's quickly recap what inequalities are. Unlike equations, which show equality between two expressions, inequalities show a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The symbols we use are: > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to).

Inequalities are used everywhere in mathematics, from basic algebra to advanced calculus. They help us define ranges and constraints, which is super useful in real-world applications. Think about situations where you have budget limits, minimum requirements, or maximum capacities – inequalities come to the rescue! So, understanding how to solve them is a crucial skill.

When solving inequalities, the main goal is similar to solving equations: isolate the variable (in this case, $x$) on one side of the inequality. However, there’s one important rule to remember: if you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. Keep this in mind as we proceed!

Initial Setup

Okay, let's get back to our problem: $-2(x+3) > x+6$. The first thing we need to do is simplify both sides of the inequality. This usually involves distributing any numbers outside parentheses and combining like terms. For our inequality, we'll start by distributing the $-2$ on the left side.

Remember, when distributing, we multiply the term outside the parentheses by each term inside the parentheses. This step helps us to remove the parentheses and make the inequality easier to work with. It’s like untangling a knot – once the parentheses are gone, we can see the individual terms more clearly and rearrange them as needed.

Step-by-Step Solution

Let's break down the solution into manageable steps.

Step 1: Distribute

We start by distributing the $-2$ across $(x+3)$:

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

$-2 * x + (-2) * 3 > x + 6$

This simplifies to:

$-2x - 6 > x + 6$

Step 2: Rearrange the Inequality

Now, we want to get all the $x$ terms on one side of the inequality and the constants on the other side. To do this, we'll add $2x$ to both sides to eliminate the $-2x$ on the left side:

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

This gives us:

$-6 > 3x + 6$

Next, we'll subtract $6$ from both sides to isolate the term with $x$:

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

Simplifying, we get:

$-12 > 3x$

Step 3: Isolate $x$

To finally solve for $x$, we need to divide both sides by $3$:

$\frac{-12}{3} > \frac{3x}{3}$

Which simplifies to:

$-4 > x$

Step 4: Rewrite the Inequality

It’s common practice to write the inequality with $x$ on the left side. So, we rewrite $-4 > x$ as:

$x < -4$

And there you have it! The solution to the inequality $-2(x+3) > x+6$ is $x < -4$.

Common Mistakes to Avoid

When solving inequalities, it’s easy to make a few common mistakes. Here are some to watch out for:

1. Forgetting to Flip the Inequality Sign: As mentioned earlier, if you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. Forgetting this is a frequent error.
2. Incorrectly Distributing: Make sure to distribute correctly by multiplying the term outside the parentheses by each term inside. A simple mistake here can throw off the entire solution.
3. Combining Unlike Terms: Always combine like terms properly. For example, you can combine $-2x$ and $x$, but you can't combine $-2x$ and $6$.

By being mindful of these common pitfalls, you can increase your accuracy and confidence in solving inequalities.

Real-World Applications

You might be wondering, “Where do inequalities actually come into play in real life?” Well, they're used in various fields, including:

• Finance: Setting budget constraints (e.g., spending less than a certain amount).
• Engineering: Determining safety margins (e.g., a bridge’s load capacity).
• Computer Science: Defining algorithm efficiency (e.g., the maximum number of steps an algorithm will take).
• Economics: Modeling supply and demand (e.g., the price range for a product).

So, the skills you're learning here aren't just for math class – they have practical value in many different areas!

Practice Problems

To solidify your understanding, let's try a couple more practice problems:

1. Solve for $x$: $3(x - 2) < 5x + 4$
2. Solve for $x$: $-4x + 7 ≥ 15$

Work through these problems step-by-step, applying the techniques we've discussed. Remember to pay close attention to the signs and to flip the inequality sign when necessary.

Answers to Practice Problems

Here are the solutions to the practice problems:

1. $3(x - 2) < 5x + 4$

   Distribute: $3x - 6 < 5x + 4$

   Rearrange: $-10 < 2x$

   Isolate $x$: $-5 < x$

   Rewrite: $x > -5$

2. $-4x + 7 ≥ 15$

   Subtract 7: $-4x ≥ 8$

   Divide by -4 (and flip the sign): $x ≤ -2$

Conclusion

Solving inequalities might seem intimidating initially, but with a clear understanding of the rules and a step-by-step approach, it becomes much more manageable. Remember to distribute correctly, combine like terms, and always flip the inequality sign when multiplying or dividing by a negative number. Guys, you've got this!

By mastering inequalities, you're not just improving your math skills – you're also gaining a valuable tool for real-world problem-solving. Keep practicing, and you’ll become a pro in no time. Now, go ahead and tackle those inequalities with confidence! If you found this helpful, feel free to share it with your friends, and let's conquer math together! Good luck, and happy solving!