Solving Inequalities: A Step-by-Step Guide

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Hey everyone! Today, we're diving into the world of inequalities, specifically how to solve a system of inequalities. Don't worry, it's not as scary as it sounds! We'll break down the process step by step, making sure you grasp every concept. Let's tackle the system:

  • 3 + x > 0
  • 2x - 7 ≤ 0

We'll go through the solution in a way that's easy to understand. So, grab your pencils and let's get started. By the end of this guide, you'll be solving systems of inequalities like a pro. This guide is designed to not only help you solve the given problem but also to provide you with a solid foundation for tackling more complex inequality problems in the future. We'll start with the basics, ensuring everyone is on the same page before moving on to more advanced techniques. Get ready to boost your math skills and gain confidence in your problem-solving abilities. Remember, practice makes perfect, so don't hesitate to work through additional examples and exercises to solidify your understanding. Let's make learning math an enjoyable and rewarding experience. This section is structured to provide clear explanations, practical examples, and helpful tips to make learning inequalities as straightforward as possible.

Understanding the Basics of Inequalities

Before we jump into the problem, let's quickly recap what inequalities are all about. Inequalities are mathematical statements that compare two expressions using symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Unlike equations, which have a single solution, inequalities often have a range of solutions. Understanding these symbols is fundamental to solving inequalities correctly. The direction of the inequality symbol indicates the relationship between the two sides. Knowing how to interpret these symbols accurately is crucial for understanding the solution sets of inequalities. Also, understanding the basic properties of inequalities is critical. These properties allow us to manipulate inequalities in a way that preserves their truth. For example, adding or subtracting the same value from both sides of an inequality doesn't change its solution set. However, multiplying or dividing both sides by a negative number does require us to flip the inequality sign. These rules are essential for solving inequalities and finding the correct solution. Let's get these fundamentals down before we move forward. This foundation will help us solve the given problem efficiently and correctly.

Inequality Symbols and Their Meanings

  • > (Greater than): Means the value on the left is larger than the value on the right.
  • < (Less than): Means the value on the left is smaller than the value on the right.
  • ≥ (Greater than or equal to): Means the value on the left is either larger than or equal to the value on the right.
  • ≤ (Less than or equal to): Means the value on the left is either smaller than or equal to the value on the right.

It's important to remember these symbols as we work through the problems. Understanding what each symbol represents is the key to interpreting the solutions we find. Each symbol has a specific meaning that directly impacts how we solve and interpret the inequality. Without a strong understanding of these symbols, it's easy to make mistakes. So, take a moment to review them. This foundational knowledge will be helpful as we begin solving our system of inequalities. Making sure you understand each symbol is a crucial step towards mastering the skill of solving inequalities. You will encounter these symbols frequently in mathematics, so it's a worthwhile investment to solidify your understanding.

Solving the First Inequality: 3 + x > 0

Alright, let's get down to business! Our first inequality is 3 + x > 0. The goal here is to isolate 'x' on one side of the inequality. To do that, we need to get rid of the '+3'. To do this, we'll subtract 3 from both sides of the inequality. This is a crucial step. When you perform the same operation on both sides, the inequality remains balanced, and the solution stays correct. Remember, the rules of algebra apply here: whatever you do to one side, you must do to the other to keep things fair. This approach ensures that we are correctly manipulating the inequality without changing its fundamental meaning. Now, let's break it down step-by-step to see how this works.

Step-by-Step Solution

  1. Original Inequality: 3 + x > 0
  2. Subtract 3 from both sides: (3 + x) - 3 > 0 - 3
  3. Simplify: x > -3

So, the solution to the first inequality is x > -3. This means that any value of 'x' that is greater than -3 will satisfy the inequality. This is a critical finding, and it means that any number larger than -3 is a valid solution. Remember, inequalities can have multiple solutions, and in this case, we have an infinite number of solutions that are all bigger than -3. Now that we've solved the first inequality, we're one step closer to solving the entire system. Understanding how to solve single inequalities is the foundation for solving systems of inequalities. So, you're doing great! Keep it up.

Solving the Second Inequality: 2x - 7 ≤ 0

Now, let's move on to the second inequality: 2x - 7 ≤ 0. Our aim is the same: isolate 'x'. This involves a couple of steps, but we'll take it slow and steady. We'll start by adding 7 to both sides, then we'll divide by 2. This process ensures that we correctly isolate 'x' while maintaining the integrity of the inequality. Each step is important, so follow along carefully. We will walk through the process with a keen eye for detail. This ensures that every operation is correct. Remember, making a mistake here could impact the final solution. Now, let's break down this process.

Step-by-Step Solution

  1. Original Inequality: 2x - 7 ≤ 0
  2. Add 7 to both sides: (2x - 7) + 7 ≤ 0 + 7
  3. Simplify: 2x ≤ 7
  4. Divide both sides by 2: (2x) / 2 ≤ 7 / 2
  5. Simplify: x ≤ 3.5

So, the solution to the second inequality is x ≤ 3.5. This means that any value of 'x' that is less than or equal to 3.5 satisfies the inequality. This result provides us with another range of values that are valid for 'x'. As we progress, we are one step closer to finding the complete solution for the system of inequalities. Now, let's see how we can put these solutions together.

Combining the Solutions: Finding the Overlap

Now we've solved each inequality individually, we need to find the values of 'x' that satisfy both inequalities. In other words, we need to find the overlap between the two solution sets. Remember, the first inequality told us that x > -3, and the second inequality told us that x ≤ 3.5. We are looking for values of 'x' that meet both these conditions. This is where a number line can come in handy, so you can visualize the overlap and make the process easier. The intersection of these two solution sets represents the final solution to the system of inequalities. We can combine our solutions by examining the range of 'x' values that meet both conditions. We will use a visual method to assist in determining the final solution.

Visualizing with a Number Line

To make this clearer, let's visualize this using a number line:

  1. Draw a number line.
  2. Mark -3 and 3.5 on the number line.
  3. For x > -3, shade the number line to the right of -3 (but don't include -3 itself because it is not included in the solution).
  4. For x ≤ 3.5, shade the number line to the left of 3.5 (including 3.5).
  5. The overlap of these two shaded regions is where both inequalities are satisfied.

The overlapping region shows where both conditions are true. This gives us the range of values that satisfy the entire system. Visualization helps us to clearly see the final solution.

Finding the Solution Set

From the number line, we can see that the solution to the system of inequalities is -3 < x ≤ 3.5. This means that 'x' can be any number greater than -3 but less than or equal to 3.5. So, the solution is the set of all real numbers that lie between -3 and 3.5, including 3.5. These numbers fulfill both inequalities. This is the final solution set, and it represents all the values of 'x' that satisfy the system of inequalities.

Writing the Solution in Interval Notation

We can also express this solution in interval notation, which is a concise way of representing the solution set. In interval notation:

  • Use parentheses () to indicate that the endpoint is not included.
  • Use square brackets [] to indicate that the endpoint is included.

So, the solution -3 < x ≤ 3.5 can be written in interval notation as (-3, 3.5]. This notation tells us that the interval starts just after -3 and includes all numbers up to and including 3.5. The interval notation is the mathematical shorthand of the solution set. It's a standard format in math and makes it easier to communicate and interpret the solution sets. Understanding interval notation is crucial for advanced math. Make sure you're comfortable with both interval notation and the number line method, as they are essential tools for solving and representing solutions to inequalities.

Conclusion: You Did It!

Congratulations, you've successfully solved the system of inequalities! You've learned how to isolate 'x' in single inequalities, visualize solutions on a number line, and express the solution in both inequality form and interval notation. Remember, the key is to take it step by step, focusing on isolating the variable and understanding the meaning of each symbol. Keep practicing, and you'll become a pro at solving inequalities in no time. Solving systems of inequalities involves breaking down each inequality, solving them individually, and then combining the solutions to find the range of values that satisfy all conditions. You now have the skills to tackle similar problems with confidence. Keep practicing and applying these concepts, and you will become proficient in solving a wide array of mathematical problems. Great job!