Solving The Inequality: X(2x + 3) < 2x(x + 4) + 10
Hey guys! Today, we're diving into a fun math problem: solving the inequality x(2x + 3) < 2x(x + 4) + 10. Inequalities might seem tricky at first, but don't worry, we'll break it down step by step. Think of it like this: we're not just looking for one answer, but a whole range of numbers that make this statement true. So, grab your thinking caps, and let's get started!
Understanding Inequalities
Before we jump into the solution, let's quickly recap what inequalities are all about. Unlike equations that have a single solution (or a few specific solutions), inequalities deal with ranges of values. The symbols we commonly use are:
- :< Less than
- :> Greater than
- :<= Less than or equal to
- :>= Greater than or equal to
When we solve an inequality, we're essentially finding all the values of 'x' that satisfy the given condition. This solution is often represented as an interval on the number line. Now, with that refresher out of the way, let's tackle our problem!
Step-by-Step Solution
Okay, let's get our hands dirty with the math! Our inequality is x(2x + 3) < 2x(x + 4) + 10. Our goal is to isolate 'x' and figure out what values make this statement true. Here’s how we can do it:
1. Expand the Expressions
First things first, let's get rid of those parentheses. We need to distribute the 'x' on the left side and the '2x' on the right side:
x(2x + 3) = 2x² + 3x
2x(x + 4) = 2x² + 8x
So, our inequality now looks like this:
2x² + 3x < 2x² + 8x + 10
2. Simplify the Inequality
Now, let's simplify things by getting all the terms to one side. We can subtract 2x² from both sides:
2x² + 3x - 2x² < 2x² + 8x + 10 - 2x²
This simplifies to:
3x < 8x + 10
Next, let's subtract 8x from both sides:
3x - 8x < 8x + 10 - 8x
This gives us:
-5x < 10
3. Isolate 'x'
Alright, we're almost there! Now we need to isolate 'x'. To do this, we'll divide both sides by -5. But here's a crucial rule to remember: when you divide or multiply an inequality by a negative number, you need to flip the inequality sign. So, let's do it:
(-5x) / -5 > 10 / -5
This gives us:
x > -2
4. Interpret the Solution
Fantastic! We've found our solution. x > -2 means that any value of 'x' greater than -2 will satisfy the original inequality. This includes numbers like -1.9, 0, 1, 10, and so on. To visualize this, we can represent it on a number line. Imagine a number line with -2 marked on it. Our solution includes all the numbers to the right of -2, but not -2 itself (since it's just “greater than,” not “greater than or equal to”).
5. Express the Solution Set
In mathematical terms, we express this solution as an interval. Since 'x' is greater than -2, the solution set is all numbers from -2 to positive infinity. We write this as:
(-2, +∞)
The parentheses indicate that -2 is not included in the solution set. If it were “greater than or equal to,” we would use a square bracket instead.
Possible Answers and the Correct Choice
Now, let's look at the possible answers you mentioned:
- A. (-∞; 2)
- B. (-∞; -2)
- C. (-2; +∞)
- D. (-1; +∞)
Based on our calculations, the correct answer is C. (-2; +∞). This interval represents all the numbers greater than -2, which is exactly what our solution x > -2 means.
Common Mistakes to Avoid
Inequalities can be a bit tricky, and it’s easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:
Forgetting to Flip the Inequality Sign
This is the most common mistake! Remember, when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. If you don’t, you’ll end up with the wrong solution.
Incorrectly Distributing
Make sure you distribute correctly when expanding expressions. Double-check that you've multiplied each term inside the parentheses by the term outside.
Misinterpreting the Solution
It's important to understand what your solution means. For example, x > -2 means all numbers greater than -2, not less than. Visualizing the solution on a number line can help prevent this mistake.
Mixing Up Interval Notation
Pay attention to whether you should use parentheses or square brackets in your interval notation. Parentheses mean the endpoint is not included, while square brackets mean it is.
Practice Makes Perfect
The best way to master inequalities is to practice! Try solving different types of inequalities, including linear, quadratic, and absolute value inequalities. The more you practice, the more comfortable you'll become with the process. You can find plenty of practice problems online or in textbooks. Work through them step by step, and don't be afraid to make mistakes – that's how you learn!
Real-World Applications of Inequalities
You might be wondering, “When will I ever use this in real life?” Well, inequalities are actually used in many practical situations. Here are a few examples:
Budgeting
When you're creating a budget, you often need to make sure your expenses are less than or equal to your income. This is an inequality! You might set up an inequality like this: expenses <= income.
Speed Limits
The speed limit on a road is a maximum speed you're allowed to travel. This can be expressed as an inequality: speed <= speed limit.
Temperature Ranges
If you're trying to keep a room at a comfortable temperature, you might set a range, like between 20°C and 25°C. This can be written as a compound inequality: 20 <= temperature <= 25.
Manufacturing
In manufacturing, companies often have tolerances for the size or weight of a product. For example, a part might need to be within a certain range of sizes to work correctly. This can be expressed using inequalities.
Fitness and Health
If you're trying to maintain a healthy weight, you might have a target calorie intake range. This can be expressed as an inequality: minimum calories <= calorie intake <= maximum calories.
Conclusion
So, there you have it! We've successfully solved the inequality x(2x + 3) < 2x(x + 4) + 10 and found that the solution set is (-2, +∞). We also covered some key concepts about inequalities, common mistakes to avoid, and real-world applications. Remember, the key to mastering inequalities is practice. Keep solving problems, and you'll become a pro in no time! If you have any more questions or want to try another problem, just let me know. Happy solving, guys!