Solving Linear Inequalities: A Quick Guide

Hey guys! Today, we're diving into the world of solving inequalities, specifically focusing on a common type you'll encounter. We'll also tackle a simple division problem that pops up along the way. So, grab your notebooks, and let's get cracking!

Understanding the Inequality: 6x + 7 ≤ -29

Alright, team, let's break down this inequality: 6x + 7 ≤ -29. Our main goal here is to isolate 'x', similar to how we solve regular equations. Think of the '≤' symbol as a balancing scale; whatever we do to one side, we must do to the other to keep it balanced. Our mission, should we choose to accept it, is to get 'x' all by itself on one side. The first step in our inequality-solving adventure is to get rid of that pesky '+7' on the left side. To do this, we perform the opposite operation, which is subtracting 7. So, we subtract 7 from both sides of the inequality. This keeps everything fair and square.

On the left side, we have 6x + 7 - 7, which simplifies beautifully to just 6x. On the right side, we have -29 - 7. Now, this is where some of you might get a little tripped up if you're not careful with your negative numbers. When you subtract 7 from -29, you're actually moving further down the number line into more negative territory. So, -29 - 7 equals -36. Our inequality now looks like this: 6x ≤ -36. See? We're one step closer to our goal! Remember, the fundamental rule of inequalities is that as long as you add or subtract the same value from both sides, the inequality sign (in this case, '≤') remains unchanged. It's like adding or removing the same weight from both sides of a real scale – it stays balanced.

The Crucial Step: Division

Now we're at the stage 6x ≤ -36. We need to get 'x' completely alone. Since 'x' is currently being multiplied by 6, the opposite operation is division. We need to divide both sides of the inequality by 6. This is a super important step, and it's where we'll encounter our second mini-problem: What is -36 divided by 6? Let's put our thinking caps on for this one. We're dealing with a negative number being divided by a positive number. Remember the rules of signs in multiplication and division: a negative divided by a positive always results in a negative. So, 36 divided by 6 is 6, and because we have a negative number involved, -36 divided by 6 is -6. That's our answer to the division question!

Applying this to our inequality, we divide both sides by 6: (6x)/6 ≤ (-36)/6. This simplifies to x ≤ -6. And there you have it! We've successfully isolated 'x'. The solution to the inequality 6x + 7 ≤ -29 is x ≤ -6. This means any number that is less than or equal to -6 will satisfy the original inequality. It's pretty neat how we can find a whole range of solutions, not just a single number. Keep practicing these steps, guys, and you'll be inequality wizards in no time! It's all about understanding the properties of inequalities and carefully applying the rules of arithmetic, especially with those tricky negative numbers.

Why the Steps Matter: Maintaining the Inequality's Truth

Let's really hammer home why each step is critical when we solve the inequality for x. We started with 6x + 7 ≤ -29. The whole point is to find all the possible values of 'x' that make this statement true. Imagine you have a number line. We want to define a boundary and say, 'Everything on this side of the boundary is a solution.' The first step, subtracting 7 from both sides, gave us 6x ≤ -36. This step is crucial because adding or subtracting any number from both sides of an inequality does not change the direction of the inequality sign. It's like adding or removing the same amount of weight from both sides of a perfectly balanced scale; it remains balanced. So, the 'less than or equal to' (≤) sign stays exactly the same. If we had messed this up, our entire solution would be wrong from the get-go.

Then we hit the division step: (6x)/6 ≤ (-36)/6, leading to x ≤ -6. Now, this is where things can get a little spicy with inequalities, guys. When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, if we had -6x ≤ 36, and we divided both sides by -6, we would get x ≥ -6 (notice the ≤ flipped to ≥). However, in our specific problem, we were dividing by a positive number (6). When you divide or multiply by a positive number, the inequality sign stays the same. So, the '≤' symbol correctly remains '≤'. This is why understanding the rules for different operations is so vital. Missing this detail means you're essentially solving a different problem. The intermediate step, 6x ≤ -36, accurately represents the state of our inequality after the first manipulation. And the final step, x ≤ -6, is the direct result of correctly dividing by 6, maintaining the integrity of the inequality. The question What is -36 divided by 6? is a critical calculation within this process, and getting it wrong means your final solution will be off. Since we're dividing a negative by a positive, the result is negative, making the answer -6. So, our final solution, x ≤ -6, is mathematically sound because we followed all the rules precisely. This means any value of x that is -6 or smaller makes the original statement true. Pretty cool, right? Always double-check those sign rules, especially when division or multiplication by negative numbers is involved!

Connecting the Dots: The Answer is -6

So, to recap, we've worked through the inequality 6x + 7 ≤ -29 step-by-step. We first isolated the term with 'x' by subtracting 7 from both sides, which correctly resulted in 6x ≤ -36. This intermediate step is crucial because it simplifies the problem while preserving the truth of the inequality. The next step involved dividing both sides by 6 to get 'x' by itself. This is where the calculation What is -36 divided by 6? comes into play. As we established, a negative number divided by a positive number yields a negative result. Therefore, -36 divided by 6 equals -6. This calculation is not just a random math problem; it's an integral part of solving the inequality. The correct result of this division, -6, is then placed on the right side of the inequality sign.

Finally, we have the solution x ≤ -6. This means that any number less than or equal to -6 is a valid solution to the original inequality. The possible answers provided were 3, -8, -6, and 9. Based on our calculations, -6 is the correct value that completes the inequality x ≤ [?]. It's essential to remember that when solving inequalities, the operations you perform matter, especially when dealing with negative numbers. Multiplying or dividing by a negative number requires flipping the inequality sign, a rule we didn't need to apply here since we divided by a positive 6. But it's a golden rule to keep in your mathematical toolkit, guys! By carefully following the rules of algebra and arithmetic, we arrive at the correct solution. It's rewarding to see how each distinct calculation, like determining -36 divided by 6, fits into the larger puzzle of solving the inequality. Keep up the great work, and don't shy away from these problems – they're building blocks for more advanced math!