Solving (3/4)(x+8) > (1/2)(2x+10) A Step By Step Guide
Hey there, math enthusiasts! Today, we're diving into the world of inequalities to solve a particularly interesting problem: What is the solution to the inequality (3/4)(x+8) > (1/2)(2x+10)? This type of question pops up frequently in algebra, and mastering it is crucial for building a solid foundation in mathematics. So, grab your thinking caps, and let's get started!
Decoding the Inequality: A Step-by-Step Approach
To effectively tackle this inequality, we'll break it down into manageable steps. Our primary goal is to isolate the variable 'x' on one side of the inequality. This will reveal the range of values for 'x' that satisfy the given condition. Let's dive in, guys!
1. Distribute the Coefficients: Clearing the Parentheses
The first thing we need to do is distribute the coefficients on both sides of the inequality. This means multiplying the fractions outside the parentheses by each term inside. For the left side, we have (3/4)(x+8). Distributing the (3/4), we get (3/4)*x + (3/4)*8. Let's simplify this: (3/4)x + 6. Remember, guys, it's all about making the equation simpler!
On the right side, we have (1/2)(2x+10). Distributing the (1/2), we get (1/2)*2x + (1/2)*10, which simplifies to x + 5. Now our inequality looks like this:
(3/4)x + 6 > x + 5
2. Gathering 'x' Terms: Isolating the Variable
Next up, we want to gather all the 'x' terms on one side of the inequality and the constants on the other. To do this, let's subtract (3/4)x from both sides. This gives us:
6 > x - (3/4)x + 5
Simplifying the right side, we need to combine x and -(3/4)x. Think of x as (4/4)x. So, (4/4)x - (3/4)x = (1/4)x. Our inequality now reads:
6 > (1/4)x + 5
3. Isolating 'x': The Final Push
Now, let's isolate the 'x' term further by subtracting 5 from both sides:
6 - 5 > (1/4)x
This simplifies to:
1 > (1/4)x
To get 'x' completely by itself, we need to get rid of the (1/4) coefficient. We can do this by multiplying both sides of the inequality by 4:
4 * 1 > 4 * (1/4)x
This gives us:
4 > x
4. The Solution Set: Interpreting the Result
So, we've arrived at the solution: 4 > x. This means that 'x' is less than 4. In interval notation, this is represented as (-\infty, 4). This is the set of all real numbers less than 4. Remember, guys, the open parenthesis indicates that 4 itself is not included in the solution set.
Putting It All Together: A Recap of the Solution
Let's quickly recap the steps we took to solve the inequality:
- Distribute the coefficients: (3/4)(x+8) > (1/2)(2x+10) became (3/4)x + 6 > x + 5.
- Gather 'x' terms: We subtracted (3/4)x from both sides to get 6 > (1/4)x + 5.
- Isolate 'x': We subtracted 5 and multiplied by 4 to get 4 > x.
- The Solution Set: We interpreted 4 > x as x < 4, which in interval notation is (-\infty, 4).
Therefore, the solution to the inequality (3/4)(x+8) > (1/2)(2x+10) is (-\infty, 4). It's like we've cracked the code, guys!
Why This Matters: The Significance of Inequalities
You might be wondering, why bother with inequalities? Well, inequalities are fundamental in many areas of mathematics and real-world applications. They help us describe situations where quantities are not necessarily equal but have a specific relationship, such as one being greater than or less than another. This comes up all the time in science, engineering, economics, and even everyday life. They are extremely powerful and beneficial mathematical tools to learn.
Real-World Examples
- Budgeting: If you have a budget of $100, you might express your spending limit as an inequality: spending <= $100.
- Speed Limits: A speed limit of 65 mph can be expressed as speed <= 65 mph.
- Manufacturing: A company might need to ensure that the weight of a product is within a certain range, expressed as an inequality.
Inequalities in Advanced Math
In higher-level mathematics, inequalities are used extensively in calculus, analysis, and optimization problems. They form the basis for concepts like limits, continuity, and the study of functions. They are also heavily used in linear programming and constraint satisfaction problems, which are essential for businesses and optimization practices.
Common Pitfalls: Avoiding Mistakes with Inequalities
Solving inequalities is similar to solving equations, but there are a few key differences to keep in mind. One of the most important is what happens when you multiply or divide both sides by a negative number.
The Negative Number Rule
When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, if we have -2x > 4, dividing both sides by -2 gives us x < -2 (notice the flipped sign!). Forgetting this rule is a common mistake, so always double-check when dealing with negative numbers.
Other Common Errors
- Incorrect Distribution: Make sure to distribute correctly, multiplying the coefficient by every term inside the parentheses.
- Sign Errors: Pay close attention to signs, especially when adding or subtracting terms across the inequality.
- Misinterpreting the Solution: Remember to correctly interpret the solution in interval notation or on a number line.
Practice Makes Perfect: Sharpening Your Skills
Like any mathematical skill, solving inequalities takes practice. The more problems you solve, the more comfortable and confident you'll become. Here are a few tips for honing your skills:
- Work through examples: Start by reviewing solved examples to understand the process.
- Practice problems: Solve a variety of problems, from simple to complex.
- Check your answers: Make sure your solution makes sense by plugging it back into the original inequality.
- Seek help when needed: Don't hesitate to ask your teacher, tutor, or classmates for help if you're stuck.
Conclusion: Mastering the Art of Inequalities
So, we've successfully navigated the world of inequalities and found the solution to (3/4)(x+8) > (1/2)(2x+10). Remember, guys, the key is to break down the problem into smaller, manageable steps, pay attention to the rules, and practice consistently. With a little effort, you'll be solving inequalities like a pro in no time!
Inequalities are more than just mathematical exercises; they are powerful tools for understanding and solving real-world problems. By mastering inequalities, you're equipping yourself with a valuable skill that will serve you well in mathematics and beyond. Keep practicing, keep exploring, and keep pushing your mathematical boundaries! You've got this! Remember, guys, math can be fun, especially when you start to master it!
And that's a wrap for today's math adventure! Keep those brains buzzing and remember, the world of mathematics is vast and exciting. There's always something new to learn and explore. So, until next time, keep solving, keep questioning, and keep that mathematical curiosity alive!