Solving Inequalities: Find Integers In A Specific Interval

Hey guys! Let's dive into some algebra today. We're gonna solve an inequality, which is super useful for all sorts of real-world problems. We'll find all the integer solutions within a specific range. So, buckle up; this is going to be fun!

Understanding the Problem: The Core Inequality

Our main task involves solving an inequality. An inequality is a mathematical statement that compares two values, indicating that they are not equal. Unlike equations that use an equals sign (=), inequalities use symbols like 'less than' (<), 'greater than' (>), 'less than or equal to' (≤), or 'greater than or equal to' (≥). In our case, we're working with the inequality: $7x - 3(4x - 8) ≤ 6x + 12 - 9x$. The objective here is to determine the values of x that satisfy this relationship. Think of it like this: We have a balance scale, and we need to find what values of x will keep the scale tipped in the 'less than or equal to' direction. The process of solving an inequality is very similar to solving an equation. We will perform algebraic manipulations to isolate the variable x on one side of the inequality. The key difference is that when we multiply or divide both sides by a negative number, we must reverse the direction of the inequality sign. Before we get into that though, it is important to understand what the notation $[4, 8]$ means. This is an interval. In this specific interval, the notation $[4, 8]$ is a closed interval, meaning it includes all real numbers between 4 and 8, including 4 and 8 themselves. If the brackets were parentheses, such as $(4, 8)$, it would mean that we exclude the end points. We have to solve the inequality and then consider only the values within this range to find our final answer. It is a bit like setting some boundaries for our solution set. Let's get started on this and get a clear understanding of the steps involved in determining the values of $x$.

First things first, let's simplify our inequality. We'll distribute the -3 across the terms inside the parentheses. This means multiplying -3 by both 4x and -8. It gives us: $7x - 12x + 24 ≤ 6x + 12 - 9x$. Now, let's combine like terms on both sides. On the left, 7x and -12x combine to -5x. On the right, 6x and -9x combine to -3x. So, our inequality now looks like: $-5x + 24 ≤ -3x + 12$. Our next step is to isolate the x terms on one side and the constants on the other side. Let's add 5x to both sides. This eliminates the -5x on the left, giving us: $24 ≤ 2x + 12$. Then, subtract 12 from both sides to get all the constants on the left: $12 ≤ 2x$. Finally, to isolate x, we divide both sides by 2: $6 ≤ x$. This is the simplified solution. It means that x must be greater than or equal to 6. But remember, we're not just looking for any x; we're interested in the x values that fall within the interval [4, 8].

Solving the Inequality Step-by-Step

Alright, let's break down the process of solving the inequality step-by-step. Remember, the inequality we're working with is: $7x - 3(4x - 8) ≤ 6x + 12 - 9x$. Our goal is to isolate x on one side of the inequality. The first thing we want to do is expand and simplify this so we have a simplified version. First, we distribute the -3 across the terms inside the parentheses: $7x - 12x + 24 ≤ 6x + 12 - 9x$. We then combine like terms on each side. On the left side, we have 7x and -12x, which combine to -5x. On the right side, 6x and -9x combine to -3x. So, now we have: $-5x + 24 ≤ -3x + 12$. Now, we want to isolate the x terms. We can add 5x to both sides: $-5x + 5x + 24 ≤ -3x + 5x + 12$. Which simplifies to: $24 ≤ 2x + 12$. Then, we subtract 12 from both sides: $24 - 12 ≤ 2x + 12 - 12$. Simplifying, we get: $12 ≤ 2x$. Finally, we divide both sides by 2 to solve for x: rac{12}{2} ≤ rac{2x}{2}. Which gives us: $6 ≤ x$. Thus, we have determined that x must be greater than or equal to 6 to satisfy the original inequality. In the context of solving inequalities, the algebraic manipulations we perform are critical to isolating the variable and determining the range of values that satisfy the inequality. Always remember to perform the same operation on both sides to maintain balance. The steps involve simplifying, combining like terms, isolating the variable, and, when necessary, reversing the inequality sign when multiplying or dividing by a negative number. This step-by-step approach not only ensures accuracy but also reinforces the underlying principles of algebraic manipulation. The skill in solving inequalities is vital for various applications, allowing for informed decision-making based on constraints and criteria. It’s like a puzzle where we're finding the pieces (the x values) that fit the rules of the game (the inequality).

Identifying Integer Solutions within the Interval

Now, let's zero in on finding the integer solutions within the specified interval of [4, 8]. We have already found that x must be greater than or equal to 6. This information provides a lower bound for our solutions. Because the interval is $[4, 8]$, it is a closed interval, which means it includes all real numbers between 4 and 8, including 4 and 8 themselves. That means our solutions must fall between 4 and 8 inclusive. Thus, we have to consider integers that satisfy both conditions: x ≥ 6 and 4 ≤ x ≤ 8. Since we need to determine the integers within the interval [4, 8], we are looking for whole numbers that are greater than or equal to 6 and also lie between 4 and 8, including 4 and 8 themselves. Let's list the integers from 4 to 8: 4, 5, 6, 7, 8. Considering the condition x ≥ 6, we select only the integers from our list that are 6 or greater. Therefore, the integers that satisfy both conditions are 6, 7, and 8. These are the solutions we are seeking. The integers 6, 7, and 8 are the only whole numbers that simultaneously satisfy the inequality and fall within the given interval. The process of identifying integer solutions involves first solving the inequality to define the range of acceptable values, then considering only the integers within the specified interval. This approach ensures accuracy and provides a clear understanding of the solution set.

Now, let's combine this knowledge. We know that x must be greater than or equal to 6, and it must also be within the interval [4, 8]. The integers that satisfy both conditions are 6, 7, and 8. If we were to substitute these values back into the original inequality, you'd find that they hold true. For example, if we let x = 6, we get: $7(6) - 3(4(6) - 8) ≤ 6(6) + 12 - 9(6)$, which simplifies to $42 - 3(24 - 8) ≤ 36 + 12 - 54$, further simplifying to $42 - 3(16) ≤ 48 - 54$, and finally, $42 - 48 ≤ -6$, which is $-6 ≤ -6$, a true statement. A similar process can be done for x = 7 and x = 8 to confirm their validity.

Conclusion: The Final Answer

So, to wrap things up, the integers that satisfy the given inequality $7x - 3(4x - 8) ≤ 6x + 12 - 9x$ within the interval [4, 8] are 6, 7, and 8. We arrived at this solution by first simplifying and solving the inequality to find the range of possible x values. Then, we considered the given interval [4, 8] and identified the integers that fell within both the solution set of the inequality and the specified interval. This process combines algebraic skills with the ability to interpret and apply mathematical constraints, providing a solid foundation for understanding and solving various types of problems. Using this step-by-step method and understanding of intervals, we can confidently identify integer solutions within specific bounds.

That's it, guys! We successfully solved the inequality and found our integer solutions. Pretty cool, huh? Keep practicing, and you'll become a pro at this. Understanding inequalities and intervals is a fundamental skill in mathematics and has a wide range of applications in various fields.