Solving 0.24x - 0.74 = -1.1: A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. It's like detective work, where we unravel the mystery of the unknown variable. Today, we'll walk through the process of solving the equation 0.24x - 0.74 = -1.1 step by step. We'll break it down in a friendly, easy-to-understand way, so you can confidently tackle similar problems in the future. Think of it as building a house: we'll lay the foundation first, then add the walls, and finally put on the roof – our solution!

Step 1: Isolating the Term with 'x'

The first thing we want to do is isolate the term that contains our variable, 'x'. In this case, that's 0.24x. The key here is to perform the same operation on both sides of the equation to maintain balance. Imagine it as a scale; if you add or subtract something on one side, you need to do the same on the other to keep it level. We have 0.24x - 0.74 = -1.1. Notice the '- 0.74' on the left side? To get rid of it, we'll do the opposite: we'll add 0.74 to both sides. Why add? Because adding 0.74 will cancel out the -0.74, leaving us with just the 0.24x term. So, let's do it: 0.24x - 0.74 + 0.74 = -1.1 + 0.74. On the left, -0.74 and +0.74 cancel each other out. On the right, we have -1.1 + 0.74. This might seem a bit tricky, but think of it as owing $1.10 and then paying back $0.74. How much do you still owe? You'd subtract 0.74 from 1.1. Doing that gives us 0.36. But remember, since we started with -1.1 (a negative number with a larger absolute value), our result will be negative. So, -1.1 + 0.74 = -0.36. Now our equation looks much simpler: 0.24x = -0.36. We've successfully isolated the term with 'x'. Feels good, right? We're one step closer to solving the mystery!

Step 2: Solving for 'x'

Now that we have 0.24x = -0.36, we're in the home stretch! We want to find out what 'x' equals, all by itself. Right now, 'x' is being multiplied by 0.24. To undo multiplication, we use division. Just like before, we'll do the same thing to both sides of the equation to keep it balanced. We're going to divide both sides by 0.24. This will cancel out the 0.24 on the left side, leaving 'x' all alone. Let's write it out: (0.24x) / 0.24 = -0.36 / 0.24. On the left, 0.24 divided by 0.24 is 1, so we're left with 1 * x, which is just x. Now, let's tackle the right side: -0.36 / 0.24. Dividing decimals can seem daunting, but it's manageable if we take it step by step. One way to make it easier is to get rid of the decimals. We can do this by multiplying both the numerator (-0.36) and the denominator (0.24) by 100. This shifts the decimal point two places to the right in both numbers, giving us -36 / 24. Now we have a simpler fraction to deal with. Both 36 and 24 are divisible by 12, so we can simplify the fraction: -36 / 12 = -3 and 24 / 12 = 2. So, -36 / 24 simplifies to -3 / 2. We can also express this as a decimal: -3 divided by 2 is -1.5. Therefore, x = -1.5. Congratulations! We've solved the equation. It might have seemed like a mountain to climb at first, but we conquered it step by step. Remember, the key to solving equations is to isolate the variable by performing inverse operations on both sides.

Step 3: Verification

Before we declare victory, it's always a good idea to double-check our answer. We can do this by plugging our solution, x = -1.5, back into the original equation: 0.24x - 0.74 = -1.1. Let's substitute -1.5 for x: 0.24 * (-1.5) - 0.74. First, we need to multiply 0.24 by -1.5. When multiplying decimals, it's often helpful to ignore the decimal points at first and multiply the numbers as if they were whole numbers. So, let's multiply 24 by 15. 24 * 15 = 360. Now, we need to figure out where the decimal point goes. In 0.24, there are two digits after the decimal point. In 1.5, there's one digit after the decimal point. So, in our result, we need a total of 2 + 1 = 3 digits after the decimal point. That means 360 becomes 0.360, or simply 0.36. But remember, we were multiplying 0.24 by -1.5, so our result is negative: -0.36. Now we can rewrite our equation with the result of the multiplication: -0.36 - 0.74. Next, we need to subtract 0.74 from -0.36. Think of this as owing $0.36 and then owing another $0.74. To find the total amount you owe, you'd add the amounts together. So, 0.36 + 0.74 = 1.10. Since we're dealing with negative numbers, our result will be negative: -1.10, or simply -1.1. Now our equation looks like this: -1.1 = -1.1. This is a true statement! It means that our solution, x = -1.5, is correct. We've successfully verified our answer. Checking your work is like putting a lock on a treasure chest – it ensures your solution is safe and sound. It's a crucial step in any mathematical problem-solving process.

A Quick Recap

Let's recap what we've done. We started with the equation 0.24x - 0.74 = -1.1. First, we isolated the term with 'x' by adding 0.74 to both sides. This gave us 0.24x = -0.36. Then, we solved for 'x' by dividing both sides by 0.24, which resulted in x = -1.5. Finally, we verified our solution by plugging it back into the original equation and confirming that it made the equation true. Each step was like a piece of a puzzle, and when we put them all together, we saw the whole picture – the solution! Remember, solving equations is a process of carefully undoing operations to isolate the variable. With practice, you'll become more comfortable and confident in your ability to tackle these problems.

Real-World Applications

You might be wondering, "Okay, this is great, but when will I ever use this in real life?" Well, the truth is, solving equations is a skill that pops up in many different areas. Think about budgeting: you might need to figure out how much you can spend on each item while staying within your budget. That often involves setting up and solving an equation. Or consider cooking: if you want to double a recipe, you'll need to multiply all the ingredients by 2. But what if you only want to make half the recipe? Then you'll need to divide the ingredients by 2, which again involves solving equations (or at least thinking through the process in an equation-like way). Even in fields like engineering and science, solving equations is a fundamental part of the job. Engineers use equations to design bridges and buildings, and scientists use them to understand the natural world. So, the skills you're learning here are not just for the classroom; they're valuable tools that can help you in many aspects of life. The more you practice, the more natural it will become to think mathematically and solve problems with confidence. Keep exploring, keep learning, and keep challenging yourself!

In conclusion, we've successfully navigated the process of solving the equation 0.24x - 0.74 = -1.1. We've isolated the variable, found the solution, and verified our answer. Remember, practice makes perfect, and the more you engage with mathematical problems, the more confident you'll become in your abilities. For further exploration and practice on equation solving, consider visiting resources like Khan Academy's Algebra I section, where you can find numerous lessons and exercises to hone your skills. Happy solving!