Solving Math Inequalities: Finding Integer Solutions

Nov 1, 2025 by Admin 53 views

Hey math enthusiasts! Ready to dive into some inequality problems? Today, we're going to tackle a few math challenges, focusing on solving inequalities and identifying the smallest integer solutions. This is super important stuff, guys, whether you're brushing up on your algebra skills or just trying to stay sharp. We'll break down the steps, making sure everything is clear and easy to follow. So, let's get started and make sure you understand how to solve inequalities with ease! This will help you be more confident in problem-solving and also improve your critical thinking skills.

Solving the Inequality 5(x + 10) - 2(x - 1) - (x + 2) >= 10

Alright, let's jump right into the first problem. We're asked to solve the inequality 5(x + 10) - 2(x - 1) - (x + 2) >= 10 and find the smallest integer solution. This might look a little intimidating at first, but trust me, it's totally manageable. We'll go step-by-step to unravel this mystery! Remember, the goal is to isolate x on one side of the inequality symbol. This involves simplifying the expression, combining like terms, and performing operations to get x alone. Pay close attention to the details – every step counts. Are you ready? Let's get started. Make sure you don't miss anything and be sure of the final answer!

First things first, we need to expand the terms. Let's multiply out the parentheses: 5 * x + 5 * 10 - 2 * x + 2 * 1 - x - 2 >= 10. This simplifies to 5x + 50 - 2x + 2 - x - 2 >= 10. Notice that we've carefully multiplied each term inside the parentheses by the number outside. This expansion is crucial for getting the correct answer. Now, we're going to combine like terms on the left side of the inequality. We have 5x, -2x, and -x. Combining these, we get 5x - 2x - x = 2x. Then, we have the constants: 50, +2, and -2. Combining these, we get 50 + 2 - 2 = 50. Thus, our inequality simplifies to 2x + 50 >= 10. Now, we need to isolate the x term. The next step is to subtract 50 from both sides of the inequality. This gives us 2x + 50 - 50 >= 10 - 50, which simplifies to 2x >= -40. Great job so far, guys! Now it's the final stage: divide both sides by 2 to solve for x. This gives us 2x / 2 >= -40 / 2, which simplifies to x >= -20. Therefore, the solution to the inequality is x >= -20. We were asked to identify the smallest integer solution. Since x must be greater than or equal to -20, the smallest integer that satisfies this is -20. Congratulations! You’ve successfully solved the inequality and found the smallest integer solution.

Solving the Inequality 8(x - 3) - (x - 2) - (2x + 5) > 5(10 - x)

Now, let's move on to the next problem! Here, we're going to solve the inequality 8(x - 3) - (x - 2) - (2x + 5) > 5(10 - x) and find the smallest integer solution. This one involves a few more steps, but don’t worry, we'll break it down just like before. The key is to stay organized and pay close attention to the signs. Remember, practice makes perfect. The more you work through these problems, the easier they will become. You will be better at it after each and every practice, and each mistake made is a new lesson learned. Let’s get to the work, shall we?

First, we need to expand all the parentheses. This means multiplying the terms outside the parentheses by each term inside. We have 8 * x - 8 * 3 - x + 2 - 2x - 5 > 5 * 10 - 5 * x. This simplifies to 8x - 24 - x + 2 - 2x - 5 > 50 - 5x. Next, we need to combine like terms on both sides of the inequality. On the left side, we have 8x, -x, and -2x. Combining these, we get 8x - x - 2x = 5x. We also have the constants -24, +2, and -5. Combining these, we get -24 + 2 - 5 = -27. So, the left side simplifies to 5x - 27. The right side already has two terms: 50 and -5x. Now, our inequality looks like this: 5x - 27 > 50 - 5x. To isolate x, we need to get all the x terms on one side and the constants on the other side. Let’s add 5x to both sides: 5x - 27 + 5x > 50 - 5x + 5x. This simplifies to 10x - 27 > 50. Now, let's add 27 to both sides: 10x - 27 + 27 > 50 + 27. This gives us 10x > 77. Finally, we divide both sides by 10 to solve for x: 10x / 10 > 77 / 10, which simplifies to x > 7.7. Thus, the solution to the inequality is x > 7.7. Because the inequality is x > 7.7, the smallest integer that satisfies this is 8. Fantastic work, team! You've successfully solved another inequality and found the smallest integer solution.

Key Concepts and Tips for Solving Inequalities

Before we wrap things up, let's recap some key concepts and tips that will help you tackle any inequality problem. Understanding these points will significantly improve your skills, guys!

Expanding Parentheses: Always start by expanding any parentheses. This often involves using the distributive property. Make sure to multiply the term outside the parentheses by every term inside. Don’t forget to pay attention to the signs! A negative sign outside the parentheses changes the signs of the terms inside.
Combining Like Terms: Simplify each side of the inequality by combining like terms. Group all the x terms together and combine the constants. This will make it easier to isolate x.
Isolating the Variable: Your primary goal is to get x by itself on one side of the inequality. Use addition, subtraction, multiplication, and division to move terms around. Remember to perform the same operation on both sides to maintain the balance.
Inequality Rules: There's one critical rule to remember: When multiplying or dividing both sides by a negative number, you must flip the direction of the inequality sign. For example, if you have -x > 5, you divide both sides by -1, and you get x < -5. This is super important! If you forget this rule, you will not get the correct solution.
Finding the Integer Solution: Once you've solved for x, determine the smallest (or largest, depending on the problem) integer that satisfies the inequality. If your solution is x > 5, the smallest integer is 6. If it’s x >= 5, the smallest integer is 5.

Practice Problems and Further Learning

Want to get even better? Practice is key! Here are some more problems you can try on your own:

Solve: 3(x - 2) + 4x - 1 >= 10 and find the smallest integer solution.
Solve: 2(x + 5) - (x - 3) < 12 and find the largest integer solution.
Solve: 4x + 7 > 2x - 3 and find the smallest integer solution.

For further learning, I highly recommend checking out the following resources:

Khan Academy: They have excellent videos and practice exercises on inequalities.
Your textbook: Work through the examples and practice problems in your math textbook.
Online math forums: Join online forums to ask questions and discuss problems with others.

Keep practicing, and you'll become a pro at solving inequalities in no time! Keep up the great work and your skills will improve, and you will learn to enjoy the challenge.

Good luck, and keep up the great work, everyone! You got this!