Solving 0.35x - 4.8 < 5.2 - 0.9x A Step-by-Step Guide

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Hey everyone! Today, we're diving into the world of inequalities and tackling a problem that might seem tricky at first glance. We're going to break down the inequality 0.35x - 4.8 < 5.2 - 0.9x step by step, making sure everyone understands how to find the solution set. So, grab your thinking caps, and let's get started!

Understanding Inequalities

Before we jump into the specific problem, let's quickly recap what inequalities are all about. Unlike equations, which have one specific solution, inequalities deal with a range of values. Think of it like this: instead of saying x equals a single number, we're saying x is greater than, less than, greater than or equal to, or less than or equal to a certain number. These concepts are foundational for understanding the problem we're about to solve.

Why are inequalities important, you ask? Well, they pop up everywhere in real life! From figuring out budget constraints to understanding speed limits, inequalities help us define boundaries and make informed decisions. In mathematics, they're crucial for solving optimization problems, analyzing functions, and much more. So, mastering inequalities is definitely worth the effort!

The Basics of Inequality Symbols

Let's quickly review the key inequality symbols:

  • < means "less than."
  • > means "greater than."
  • ≤ means "less than or equal to."
  • ≥ means "greater than or equal to."

These symbols are the building blocks of inequalities, and understanding them is crucial for interpreting and solving problems. Remember, the "open" side of the symbol always faces the larger value.

Properties of Inequalities

Now, let's talk about the rules we need to follow when manipulating inequalities. Most of the rules are similar to those for equations, but there's one crucial difference:

  1. Addition and Subtraction: You can add or subtract the same number from both sides of an inequality without changing its direction. This is just like equations!
  2. Multiplication and Division by a Positive Number: You can multiply or divide both sides of an inequality by the same positive number without changing its direction. Again, similar to equations.
  3. Multiplication and Division by a Negative Number: This is the key difference! When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. This is super important to remember!

Understanding these properties is essential for correctly solving inequalities. Forgetting to flip the inequality sign when multiplying or dividing by a negative number is a common mistake, so always double-check!

Solving the Inequality 0.35x - 4.8 < 5.2 - 0.9x

Okay, now that we've got the basics down, let's tackle our specific problem: 0.35x - 4.8 < 5.2 - 0.9x. We're going to solve this inequality step by step, just like we would solve an equation, but keeping in mind the rules we just discussed.

Step 1: Combine the 'x' terms

Our first goal is to get all the x terms on one side of the inequality. To do this, we can add 0.9x to both sides. This will eliminate the -0.9x term on the right side:

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

This simplifies to:

1.25x - 4.8 < 5.2

Great! We've successfully combined the x terms. Now, let's move on to the next step.

Step 2: Isolate the 'x' term

Next, we want to isolate the x term by getting rid of the -4.8 on the left side. We can do this by adding 4.8 to both sides:

1.25x - 4.8 + 4.8 < 5.2 + 4.8

This simplifies to:

1.25x < 10

We're getting closer! Now, we just need to get x by itself.

Step 3: Solve for 'x'

To solve for x, 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:

1.25x / 1.25 < 10 / 1.25

This simplifies to:

x < 8

Woo-hoo! We've solved the inequality! This means that any value of x that is less than 8 will satisfy the original inequality.

Expressing the Solution Set in Interval Notation

Now that we've found the solution, let's express it in interval notation. Interval notation is a way of writing sets of numbers using intervals. It's a concise and clear way to represent the solution set of an inequality.

Understanding Interval Notation

Here's a quick rundown of the symbols used in interval notation:

  • ( ) Parentheses indicate that the endpoint is not included in the interval. This is used for strict inequalities like < and >. For example, (a, b) means all numbers between a and b, but not including a and b.
  • [ ] Square brackets indicate that the endpoint is included in the interval. This is used for inequalities like ≤ and ≥. For example, [a, b] means all numbers between a and b, including a and b.
  • ∞ Infinity symbol represents a never-ending range. We always use parentheses with infinity because infinity is not a specific number.
  • -∞ Negative infinity symbol represents a never-ending range in the negative direction. Again, we always use parentheses with negative infinity.

Applying Interval Notation to Our Solution

Our solution to the inequality is x < 8. This means all numbers less than 8, but not including 8 itself. In interval notation, this is written as:

(-∞, 8)

Why (-∞, 8)?

  • -∞ represents the fact that the solution extends infinitely in the negative direction.
  • 8 is the upper bound of the solution set.
  • The parenthesis ) indicates that 8 is not included in the solution set because our inequality is strictly less than (<).

Connecting the Solution to the Options

Now, let's take a look at the options provided in the original problem:

A. (-∞, -8) B. (-∞, 8) C. (-8, ∞) D. (8, ∞)

Comparing our solution (-∞, 8) to the options, we can see that the correct answer is:

B. (-∞, 8)

Visualizing the Solution on a Number Line

Another helpful way to understand the solution set is to visualize it on a number line. This can make it even clearer which values of x satisfy the inequality.

Drawing the Number Line

  1. Draw a horizontal line. This represents the number line.
  2. Mark the key value, which in our case is 8, on the number line.
  3. Since our inequality is x < 8, we use an open circle at 8 to indicate that it's not included in the solution set.
  4. Shade the region to the left of 8, representing all numbers less than 8. This shaded region is our solution set.
  5. Draw an arrow extending to the left, indicating that the solution set continues to negative infinity.

By looking at the number line, you can easily see that all the numbers to the left of 8 are solutions to the inequality.

Number Line and Interval Notation

The number line representation directly corresponds to the interval notation. The shaded region extending from negative infinity up to, but not including, 8 is precisely what the interval (-∞, 8) represents.

Common Mistakes to Avoid

Solving inequalities can be tricky, and there are a few common mistakes that students often make. Let's go over them so you can avoid them!

  1. Forgetting to Flip the Inequality Sign: As we mentioned earlier, this is the most common mistake. Remember, when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.
  2. Incorrectly Applying Interval Notation: Make sure you understand the difference between parentheses and square brackets. Use parentheses ( ) when the endpoint is not included (strict inequalities) and square brackets [ ] when the endpoint is included (inequalities with "or equal to").
  3. Misinterpreting the Inequality Symbol: Double-check which direction the inequality symbol is pointing. It's easy to mix up < and > if you're not careful.
  4. Arithmetic Errors: Simple arithmetic mistakes can throw off your entire solution. Take your time and double-check your calculations.

By being aware of these common pitfalls, you can significantly improve your accuracy when solving inequalities.

Real-World Applications of Inequalities

We've talked about the math behind inequalities, but let's take a moment to appreciate how they're used in the real world. Inequalities aren't just abstract concepts; they're powerful tools for solving practical problems.

  1. Budgeting: Inequalities can help you determine how much you can spend while staying within your budget. For example, if you have a monthly budget of $100 for entertainment, you can use an inequality to represent the amount you can spend each week.
  2. Travel Planning: Inequalities can help you figure out how far you can drive on a tank of gas or how much time you'll need to reach your destination at a certain speed.
  3. Health and Fitness: Inequalities can be used to set goals for your workouts or track your calorie intake. For example, you might use an inequality to represent the number of calories you need to burn to lose weight.
  4. Engineering and Science: Inequalities are used extensively in these fields to model constraints, optimize processes, and ensure safety.

These are just a few examples, but they illustrate how inequalities are a fundamental part of problem-solving in many different areas of life.

Practice Problems

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

  1. Solve the inequality: 2x + 3 > 7
  2. Solve the inequality: -3x - 5 ≤ 10

Try solving these on your own, and then check your answers. Remember to pay attention to the rules for flipping the inequality sign!

Conclusion

Guys, we've covered a lot today! We've explored the basics of inequalities, learned how to solve them step by step, expressed solutions in interval notation, visualized them on a number line, and discussed common mistakes to avoid. We've also seen how inequalities are used in the real world.

Solving inequalities is a fundamental skill in mathematics, and mastering it will open doors to more advanced concepts. So, keep practicing, stay curious, and don't be afraid to tackle challenging problems!

If you have any questions or want to explore more advanced topics, feel free to reach out. Happy solving!