Solving The Inequality (x+2)(x-1)+26 < (x+4)(x+5) A Step-by-Step Guide

Hey guys! Let's dive into solving this interesting inequality problem: (x+2)(x-1)+26 < (x+4)(x+5). It might look a bit intimidating at first, but trust me, we'll break it down step-by-step and make it super easy to understand. Our main goal here is to find the range of values for 'x' that make this inequality true. So, grab your thinking caps, and let's get started!

Understanding Inequalities

Before we jump into the nitty-gritty of solving this specific problem, let's quickly recap what inequalities are all about. Think of an inequality as a statement that compares two expressions, but instead of saying they are equal, it says one is greater than, less than, greater than or equal to, or less than or equal to the other. We use symbols like '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to) to express these relationships. When we solve an inequality, we're essentially finding the set of all values that satisfy the comparison.

The key difference between solving equations and inequalities is how certain operations affect the direction of the inequality. For instance, multiplying or dividing both sides by a negative number flips the inequality sign. This is a crucial point to remember as we work through our problem. Inequalities pop up everywhere in math and real-world applications. From determining the range of acceptable values in engineering to figuring out financial constraints, understanding inequalities is a super valuable skill.

Now, why are inequalities so important? Well, in the real world, things aren't always perfectly equal. Sometimes we need to deal with situations where one thing is more than or less than another. Imagine you're planning a budget: you need to make sure your expenses are less than or equal to your income. Or think about setting temperature limits in a laboratory: you might need to keep the temperature within a certain range. Inequalities give us the tools to model and solve these kinds of problems. They allow us to define boundaries, set limits, and make informed decisions based on constraints. So, mastering inequalities isn't just about acing your math test; it's about equipping yourself with a powerful problem-solving technique that you can apply in countless situations.

Step-by-Step Solution

Okay, let's get our hands dirty with the actual problem. We need to solve the inequality (x+2)(x-1)+26 < (x+4)(x+5). The first thing we need to do is expand both sides of the inequality. This means multiplying out the brackets to get rid of those pesky parentheses. On the left side, we have (x+2)(x-1). If we use the distributive property (or the FOIL method), we get x*(x-1) + 2*(x-1), which simplifies to x^2 - x + 2x - 2. Combining like terms, the left side becomes x^2 + x - 2. Don't forget that +26 lurking at the end! So, the entire left side simplifies to x^2 + x - 2 + 26, which is x^2 + x + 24. Now, let's tackle the right side. We have (x+4)(x+5). Again, using the distributive property, we get x*(x+5) + 4*(x+5), which simplifies to x^2 + 5x + 4x + 20. Combining like terms, the right side becomes x^2 + 9x + 20. Awesome! We've expanded both sides.

Now that we've expanded both sides, our inequality looks like this: x^2 + x + 24 < x^2 + 9x + 20. The next step is to simplify the inequality by getting all the terms on one side. We want to isolate 'x' and figure out its possible values. Notice that we have an x^2 term on both sides. If we subtract x^2 from both sides, these terms cancel each other out. This leaves us with x + 24 < 9x + 20. Much simpler, right? Next, let's get all the 'x' terms on one side and the constants on the other. We can subtract 'x' from both sides, which gives us 24 < 8x + 20. Now, let's subtract 20 from both sides to isolate the 'x' term further. This gives us 4 < 8x. We're almost there!

To finally solve for 'x', we need to get 'x' all by itself. We have 4 < 8x. This means 8 times 'x' is greater than 4. To find 'x', we simply divide both sides of the inequality by 8. Remember, since we're dividing by a positive number, the inequality sign stays the same. So, we have 4/8 < x, which simplifies to 1/2 < x. Ta-da! We've solved the inequality. This means that 'x' must be greater than 1/2 for the original inequality to hold true. We can also write this as x > 1/2. This is our solution. It's super important to double-check your answer. You can do this by picking a value for 'x' that's greater than 1/2 and plugging it back into the original inequality to make sure it works. For example, let's try x = 1. Plugging this into (x+2)(x-1)+26 < (x+4)(x+5), we get (1+2)(1-1)+26 < (1+4)(1+5), which simplifies to 26 < 30. This is true! So, our solution seems correct. Great job!

Visualizing the Solution

Sometimes, it's super helpful to visualize the solution to an inequality. This can make it easier to understand what the solution actually means. One of the best ways to visualize the solution is by using a number line. A number line is just a straight line where we can represent numbers. We mark the important points, like our solution x > 1/2, and then shade the regions that satisfy the inequality. So, let's draw a number line. We'll mark the point 1/2 on the line. Now, since our solution is x > 1/2, this means we want all the numbers that are greater than 1/2. On the number line, these are all the numbers to the right of 1/2. We'll shade the region to the right of 1/2 to represent our solution.

There's one more little detail we need to think about: Do we include 1/2 in our solution? Since our inequality is strictly 'greater than' (x > 1/2), we don't include 1/2 itself. To show this on the number line, we use an open circle at 1/2. If the inequality was 'greater than or equal to' (x ≥ 1/2), we would use a closed circle to indicate that 1/2 is included. Visualizing the solution on a number line can be especially useful when you have more complex inequalities, or when you're dealing with systems of inequalities (where you have multiple inequalities that need to be satisfied at the same time). It gives you a clear picture of the range of values that work.

Another way to think about visualizing solutions is in the context of real-world problems. Imagine 'x' represents the number of hours you can work per week. If the solution to an inequality is x > 1/2, it means you need to work more than half an hour to meet a certain requirement. The number line helps you see this range of possible hours at a glance. Visualizing solutions helps bridge the gap between abstract math and concrete situations, making the concepts more meaningful and easier to remember.

Common Mistakes to Avoid

Alright, now that we've successfully solved the inequality, let's talk about some common pitfalls that students often encounter. Avoiding these mistakes can save you a lot of headaches and ensure you get the correct answer. One of the most frequent errors is forgetting to flip the inequality sign when you multiply or divide both sides by a negative number. This is a crucial rule, and it's super easy to overlook. Remember, if you're multiplying or dividing by a negative, the direction of the inequality changes. For example, if you have -2x < 4, you need to divide both sides by -2, which means the inequality becomes x > -2.

Another common mistake happens during the expansion and simplification steps. It's easy to make arithmetic errors when you're multiplying out brackets or combining like terms. This is why it's super important to double-check your work at each step. Write out each step clearly and carefully, and don't try to rush through the process. A small mistake in the early stages can throw off your entire solution. Another pitfall is misinterpreting the solution. For instance, if you get x > 1/2, it means 'x' can be any number greater than 1/2, but not 1/2 itself. Make sure you understand what the inequality symbol means and how it translates into the set of possible solutions. When you're visualizing the solution on a number line, remember to use an open circle for strict inequalities (like > or <) and a closed circle for inequalities that include equality (like ≥ or ≤).

Finally, always, always, always check your solution! Pick a value within the range you found and plug it back into the original inequality to make sure it holds true. This is the best way to catch any mistakes you might have made along the way. For example, if we solved an inequality and got x > 3, we could plug in x = 4 into the original inequality and verify that it works. Making these checks a habit can significantly improve your accuracy. By being aware of these common mistakes and taking steps to avoid them, you'll become a pro at solving inequalities in no time!

Real-World Applications

You might be wondering,