Solving Inequalities Finding The Solution Set To 0.35x - 4.8 < 5.2 - 0.9x

Hey there, math enthusiasts! Today, we're diving into the world of linear inequalities, specifically tackling the problem of finding the solution set for the inequality $0.35x - 4.8 < 5.2 - 0.9x$. This type of problem is a staple in algebra, and mastering it will not only help you ace your exams but also sharpen your problem-solving skills in general. We'll break down the steps, explain the reasoning behind each, and make sure you're confident in tackling similar problems. So, let's get started!

Understanding Linear Inequalities

Before we jump into the specifics of our problem, let's take a moment to understand what linear inequalities are all about. Simply put, a linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike linear equations, which have a single solution (or a finite set of solutions), linear inequalities often have a range of solutions, forming an interval on the number line. This interval represents all the values of the variable that satisfy the inequality.

Why are linear inequalities important? They pop up everywhere in real-world applications, from determining budget constraints to modeling physical phenomena. For instance, you might use a linear inequality to figure out how many items you can buy within a certain budget or to describe the range of temperatures in which a particular chemical reaction can occur. So, understanding how to solve them is a valuable skill.

When solving linear inequalities, our goal is to isolate the variable on one side of the inequality symbol. We do this by performing the same operations on both sides, just like we do with equations. However, there's one crucial difference: when we multiply or divide both sides by a negative number, we must flip the direction of the inequality symbol. This is because multiplying or dividing by a negative number reverses the order of the number line. For example, if 2 < 3, then multiplying both sides by -1 gives -2 > -3.

Now that we have a solid grasp of the fundamentals, let's get our hands dirty and solve the inequality at hand.

Step-by-Step Solution to $0.35x - 4.8 < 5.2 - 0.9x$

Our mission is to find the interval that represents all the possible values of x that make the inequality $0.35x - 4.8 < 5.2 - 0.9x$ true. To do this, we'll follow a series of algebraic steps, keeping in mind the golden rule of flipping the inequality sign when multiplying or dividing by a negative number.

1. Combining Like Terms: The First Step to Isolation

The first thing we want to do is gather all the x terms on one side of the inequality and the constant terms on the other. This makes the inequality cleaner and easier to work with. To achieve this, we can add $0.9x$ to both sides of the inequality. This will eliminate the $-0.9x$ term on the right side and bring the x terms together on the left side. We get:

$0.35x - 4.8 + 0.9x < 5.2 - 0.9x + 0.9x$

Simplifying this gives us:

$1.25x - 4.8 < 5.2$

Now, let's move the constant terms to the right side. We can do this by adding 4.8 to both sides of the inequality:

$1.25x - 4.8 + 4.8 < 5.2 + 4.8$

This simplifies to:

$1.25x < 10$

2. Isolating x: The Final Push

We're almost there! Now we have all the x terms on one side and the constants on the other. The final step is to isolate x completely. Currently, x is being multiplied by 1.25. To undo this multiplication, we need to divide both sides of the inequality by 1.25. Since 1.25 is a positive number, we don't need to worry about flipping the inequality sign.

So, we divide both sides by 1.25:

rac{1.25x}{1.25} < rac{10}{1.25}

This simplifies to:

$x < 8$

3. The Solution Set: Interpreting the Result

We've arrived at the solution! The inequality $x < 8$ tells us that any value of x that is less than 8 will satisfy the original inequality. This is an infinite set of numbers, and we can represent it using interval notation. In interval notation, we use parentheses to indicate that the endpoint is not included in the interval and brackets to indicate that the endpoint is included. Since x is strictly less than 8, we use a parenthesis.

So, the solution set in interval notation is $(-\infty, 8)$. This means that all numbers from negative infinity up to (but not including) 8 are solutions to the inequality.

Connecting the Solution to the Options

Now that we've found the solution set, let's revisit the options provided in the problem and see which one matches our result.

The options were:

A. $(-\infty, -8)$ B. $(-\infty, 8)$ C. $(-8, \infty)$ D. $(8, \infty)$

As we can clearly see, option B. $(-\infty, 8)$ matches our solution set perfectly. So, that's the correct answer!

Common Pitfalls and How to Avoid Them

Solving linear inequalities is a straightforward process, but there are a few common mistakes that students often make. Let's discuss these pitfalls and how to steer clear of them.

1. Forgetting to Flip the Inequality Sign

The most common mistake is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. This can completely change the solution set. Always remember this crucial rule!

How to avoid it: Make it a habit to explicitly check if you're multiplying or dividing by a negative number. If you are, flip the sign immediately. You can even circle the inequality sign as a reminder.

2. Incorrectly Combining Like Terms

Another common mistake is messing up the arithmetic when combining like terms. This can lead to an incorrect inequality and, consequently, the wrong solution.

How to avoid it: Take your time and double-check your calculations. Write out each step clearly and carefully. If you're prone to errors, use a calculator to verify your arithmetic.

3. Misinterpreting Interval Notation

Understanding interval notation is essential for expressing the solution set correctly. Confusing parentheses and brackets can lead to an incorrect answer.

How to avoid it: Remember that parentheses indicate that the endpoint is not included, while brackets indicate that it is. Pay close attention to the inequality symbol. If it's < or >, use parentheses. If it's ≤ or ≥, use brackets. Draw a number line and visualize the interval if it helps.

4. Not Checking the Solution

It's always a good idea to check your solution to make sure it's correct. This can help you catch any errors you might have made along the way.

How to avoid it: Pick a value within your solution interval and plug it back into the original inequality. If the inequality holds true, your solution is likely correct. You can also pick a value outside the interval to confirm that it doesn't satisfy the inequality.

Real-World Applications of Linear Inequalities

As we mentioned earlier, linear inequalities have numerous applications in the real world. Let's explore a few examples to see how they're used in different contexts.

1. Budgeting and Finance

Imagine you have a budget of $100 for groceries. You want to buy some fruits that cost $2 per pound and some vegetables that cost $1.50 per pound. You can use a linear inequality to determine the possible combinations of fruits and vegetables you can buy without exceeding your budget.

Let x represent the number of pounds of fruits and y represent the number of pounds of vegetables. The inequality would be:

$2x + 1.50y ≤ 100$

Solving this inequality (or graphing it) will give you the range of possible values for x and y that satisfy your budget constraint.

2. Manufacturing and Production

In manufacturing, linear inequalities can be used to model production constraints. For example, a factory might have a limited amount of raw materials or a limited number of working hours. These constraints can be expressed as linear inequalities.

Suppose a factory produces two types of products, A and B. Product A requires 2 hours of labor and 3 units of raw material, while product B requires 1 hour of labor and 4 units of raw material. If the factory has a maximum of 100 hours of labor and 120 units of raw material available, we can set up a system of linear inequalities to represent these constraints:

$2x + y ≤ 100$ (labor constraint)

$3x + 4y ≤ 120$ (raw material constraint)

where x is the number of units of product A and y is the number of units of product B. Solving this system of inequalities will help the factory determine the optimal production levels for each product.

3. Health and Nutrition

Linear inequalities can also be used in health and nutrition to model dietary requirements. For instance, a person might need to consume a certain number of calories, protein, and carbohydrates each day.

Let's say a person needs to consume at least 2000 calories per day. They plan to eat a combination of foods, each with a different calorie content. We can use a linear inequality to model this requirement.

If food A has 300 calories per serving and food B has 500 calories per serving, the inequality would be:

$300x + 500y ≥ 2000$

where x is the number of servings of food A and y is the number of servings of food B. This inequality will help the person determine the possible combinations of foods they can eat to meet their calorie needs.

Conclusion: Mastering Linear Inequalities

Congratulations, guys! You've made it to the end of our comprehensive guide on solving linear inequalities. We've covered the basics, walked through a step-by-step solution, discussed common pitfalls, and explored real-world applications. By now, you should have a solid understanding of how to tackle these types of problems with confidence.

Remember, the key to mastering any math concept is practice. So, keep working on problems, reviewing the concepts, and asking questions when you're unsure. With consistent effort, you'll become a pro at solving linear inequalities and many other mathematical challenges. Keep up the great work!