It's a common scenario in math class, isn't it? You're faced with an inequality, and the goal isn't just to find a solution, but the biggest whole number that fits. Let's dive into one such puzzle: finding the largest integer solution for the inequality x - 5 > 4x - 1.
At first glance, it might seem a bit daunting, especially with the variables on both sides. But the beauty of algebra is that it gives us a systematic way to untangle these problems. The core idea here is to isolate 'x' and then figure out what that tells us about the possible values of 'x'.
So, how do we start? We want to get all the 'x' terms on one side and the constant numbers on the other. A good first step is to subtract 'x' from both sides. This gives us: -5 > 3x - 1.
Next, let's move the constant term. We can add 1 to both sides: -5 + 1 > 3x, which simplifies to -4 > 3x.
Now, to get 'x' by itself, we need to divide both sides by 3. Here's a crucial point in solving inequalities: when you divide or multiply by a negative number, you have to flip the inequality sign. But since we're dividing by a positive number (3), the sign stays the same. So, we get: -4/3 > x, or more commonly written as x < -4/3.
What does 'x < -4/3' actually mean? It means 'x' must be any number that is strictly less than negative four-thirds. If we think about this on a number line, -4/3 is approximately -1.33. So, any number smaller than that will satisfy the inequality.
Now for the fun part: finding the largest integer that fits this condition. Integers are whole numbers (positive, negative, or zero). We're looking for the biggest whole number that is less than -1.33. Let's consider some integers around that value:
- Is 0 less than -1.33? No.
- Is -1 less than -1.33? No, -1 is actually greater than -1.33.
- Is -2 less than -1.33? Yes, it is!
Since -2 is less than -1.33, and any integer smaller than -2 (like -3, -4, and so on) also satisfies the condition, -2 is indeed the largest whole number that is less than -1.33. It's the closest integer to -1.33 on the left side of the number line.
So, the maximum integer solution to the inequality x - 5 > 4x - 1 is -2. It's a neat little journey from a jumble of variables to a clear, single integer answer, isn't it?
