Solving Inequalities: Examples And Solutions

Hey there, math enthusiasts! Let's dive into the world of inequalities. Don't worry, it's not as scary as it sounds. We'll break down the process step by step, using the examples you provided. Ready to crack these problems? Let's go!

Solving the First Inequality: 4х-12*2х+32≥0

Alright, first things first, we've got the inequality: 4x - 12 * 2x + 32 ≥ 0. Our goal here is to find the values of 'x' that make this statement true. To do that, we need to simplify and isolate 'x'. This might seem a bit daunting at first, but trust me, with a few simple steps, we'll get there. Solving inequalities is a fundamental skill in algebra, and it opens the door to understanding a vast range of mathematical concepts. Understanding how to manipulate and solve inequalities is crucial for everything from basic problem-solving to more complex applications in fields like physics, economics, and computer science. Think of it as building a strong foundation for your mathematical journey. Let's start with simplifying our inequality. We've got 4x - 12 * 2x + 32 ≥ 0. Notice the multiplication first: 12 * 2x = 24x. So our inequality becomes: 4x - 24x + 32 ≥ 0. Now, let's combine the 'x' terms: 4x - 24x = -20x. This gives us: -20x + 32 ≥ 0. See? We're already making progress. This type of inequality is linear, meaning the highest power of 'x' is 1. The approach to solving a linear inequality mirrors the steps you'd use for a linear equation, but with a crucial twist: when you multiply or divide both sides by a negative number, you must flip the inequality sign. Remember this key rule as it is very crucial! Now that we have a simplified expression, let's focus on isolating 'x'. Think of it as playing a game of hide-and-seek; our aim is to get 'x' all by itself on one side of the inequality. To do this, we'll first subtract 32 from both sides of the inequality: -20x + 32 - 32 ≥ 0 - 32, which simplifies to -20x ≥ -32. Next, we divide both sides by -20. But hey, remember the rule about flipping the inequality sign? Since we're dividing by a negative number, we must flip the ≥ to ≤. So, we get x ≤ (-32) / (-20). Let's simplify that fraction. -32 / -20 = 8/5, or 1.6. Therefore, the solution to our first inequality is x ≤ 1.6. This means any value of 'x' that is less than or equal to 1.6 will satisfy the original inequality. Pretty cool, huh? The ability to solve this type of inequality allows us to determine the range of values that will keep an equation true or false, a skill applicable to a variety of real-world scenarios, from calculating the maximum load a bridge can support to determining the optimal price for a product.

Step-by-Step Breakdown

• Original inequality: 4x - 12 * 2x + 32 ≥ 0
• Simplify: 4x - 24x + 32 ≥ 0
• Combine 'x' terms: -20x + 32 ≥ 0
• Subtract 32 from both sides: -20x ≥ -32
• Divide by -20 (and flip the inequality sign): x ≤ 1.6
• Solution: x ≤ 1.6

Solving the Second Inequality: 3^(2x+1) - 28 * 3^x + 9 ≤ 0

Alright, let's roll up our sleeves and tackle this exponential inequality: 3^(2x+1) - 28 * 3^x + 9 ≤ 0. Now, this one looks a bit different, doesn't it? But don't worry; we'll break it down step by step. The key to this problem lies in recognizing patterns and making smart substitutions. Remember that solving inequalities is an essential skill that helps us in understanding the range of solutions within different constraints, offering us insights into numerous real-world applications. Exponential inequalities can model all sorts of scenarios, from population growth to radioactive decay, and mastering them unlocks a deeper understanding of these processes. The first step we can take is to rewrite the expression 3^(2x+1). Using exponent rules, we can rewrite this as 3^(2x) * 3^1. We can also rewrite 3^(2x) as (3^x)2. This is a very useful trick and can often simplify the process of solving such equations. So, our inequality becomes: 3 * (3^x)2 - 28 * 3^x + 9 ≤ 0. Now, here's where the magic happens. Let's make a substitution to make this easier to handle. Let's say y = 3^x. If we substitute 'y' into our inequality, we get: 3y^2 - 28y + 9 ≤ 0. This is a quadratic inequality, which is much easier to work with than the original exponential form. Now we're dealing with a quadratic inequality. The next step is to solve this quadratic inequality, and we can do that by factoring or using the quadratic formula to find the roots of the corresponding quadratic equation (3y^2 - 28y + 9 = 0). Factoring works here, so let's use that. We are looking for two numbers that multiply to give (3 * 9) = 27 and add to give -28. Those numbers are -1 and -27. So, we rewrite the middle term of the quadratic expression using these numbers: 3y^2 - 27y - y + 9 ≤ 0. Then, we can factor by grouping: 3y(y - 9) - 1(y - 9) ≤ 0, which further simplifies to (3y - 1)(y - 9) ≤ 0. Now we know the critical points, where the expression equals zero. We get y = 1/3 and y = 9. These are the values where the inequality changes its sign. To find the solution, we'll test intervals. We have three intervals to consider: y < 1/3, 1/3 < y < 9, and y > 9. Test a value from each interval in the inequality (3y - 1)(y - 9) ≤ 0. For y < 1/3, let's try y = 0. (3(0) - 1)(0 - 9) = (-1)(-9) = 9, which is not less than or equal to 0. So, this interval is not a solution. For 1/3 < y < 9, let's try y = 1. (3(1) - 1)(1 - 9) = (2)(-8) = -16, which is less than or equal to 0. So, this interval is part of our solution. For y > 9, let's try y = 10. (3(10) - 1)(10 - 9) = (29)(1) = 29, which is not less than or equal to 0. So, this interval is not a solution. So the solution in terms of 'y' is 1/3 ≤ y ≤ 9. But remember, we need to find the solution for 'x'. We know that y = 3^x. Substitute back: 1/3 ≤ 3^x ≤ 9. Now, we can rewrite the inequality in terms of the same base (3). 1/3 is the same as 3^(-1), and 9 is the same as 3^2. So we have: 3^(-1) ≤ 3^x ≤ 3^2. Since the base is greater than 1, we can simply compare the exponents: -1 ≤ x ≤ 2. And there you have it! The solution to the inequality 3^(2x+1) - 28 * 3^x + 9 ≤ 0 is -1 ≤ x ≤ 2. It’s important to practice these techniques and not get discouraged if it seems tough at first. Remember that, with practice, these skills become second nature. Understanding how to solve these problems can drastically improve your problem-solving capabilities in math, physics, and many other fields.

Step-by-Step Breakdown

• Original inequality: 3^(2x+1) - 28 * 3^x + 9 ≤ 0
• Rewrite using exponent rules: 3 * (3^x)2 - 28 * 3^x + 9 ≤ 0
• Substitute y = 3^x: 3y^2 - 28y + 9 ≤ 0
• Factor the quadratic: (3y - 1)(y - 9) ≤ 0
• Find the roots: y = 1/3 and y = 9
• Determine the interval where the inequality is true: 1/3 ≤ y ≤ 9
• Substitute back 3^x for y: 1/3 ≤ 3^x ≤ 9
• Rewrite with the same base: 3^(-1) ≤ 3^x ≤ 3^2
• Solve for x: -1 ≤ x ≤ 2
• Solution: -1 ≤ x ≤ 2

Conclusion: Keep Practicing!

Alright, guys, we've made it through both inequalities. You've seen how to solve a linear inequality and an exponential one. Remember, practice is key. The more you work through these types of problems, the more comfortable and confident you'll become. Don't hesitate to go back and review the steps, and most importantly, keep that curiosity burning. Math can be fun and rewarding, and solving inequalities is just one of many exciting adventures you can embark on. Keep practicing and keep learning! You've got this!