Solving Inequalities: A Step-by-Step Guide

Hey guys! Let's dive into the world of inequalities and learn how to solve them. In this guide, we'll break down the process step-by-step, making it super easy to understand. We'll be tackling the inequality: $\bf{9.5 \leq -2(x - 3.25)}$. Don't worry if it looks a little intimidating at first; we'll break it down piece by piece. Understanding inequalities is a fundamental skill in mathematics, popping up everywhere from basic algebra to advanced calculus. So, grabbing a solid grasp of this concept is definitely worth your time. Let's get started and make this journey a breeze!

Step 1: Distribute the -2

Alright, first things first, we need to get rid of those parentheses. To do this, we'll distribute the $\bf{-2}$ across the terms inside the parentheses. Remember, distributing means multiplying each term inside the parentheses by the number outside. So, we'll multiply $-2$ by $x$ and $-2$ by $-3.25$. This gives us:

$\bf{9.5 \leq -2x + 6.5}$

See? Not so bad, right? We've simplified the right side of the inequality, and now it looks a bit cleaner. It's like we're tidying up our mathematical workspace before we start the main work. The distribution step is crucial because it sets the stage for isolating the variable, which is our ultimate goal. It transforms the expression into a more manageable form, allowing us to move forward systematically. Pay close attention to the signs – a small mistake here can lead to a completely different answer. Double-checking your multiplication, especially when dealing with negative numbers, can save you a lot of headaches later on. Always remember that a negative times a negative equals a positive. And always keep the inequality sign as is during the distribution process, which allows us to get started with the isolating process. This first step is the cornerstone upon which the rest of our solution is built. Getting this part right will save you a lot of trouble!

Step 2: Isolate the Variable Term

Our next move is to get the term with the variable (the $-2x$ term) by itself on one side of the inequality. To do this, we need to get rid of that $+6.5$. We do this by subtracting $\bf{6.5}$ from both sides of the inequality. Remember, whatever we do to one side, we must do to the other to keep things balanced.

So, subtracting $6.5$ from both sides, we get:

$\bf{9.5 - 6.5 \leq -2x + 6.5 - 6.5}$

This simplifies to:

$\bf{3 \leq -2x}$

We're making good progress! Now, we've isolated the term with our variable, making it easier to solve for $x$. This step is all about maintaining the equality (or inequality, in this case). Think of it like a seesaw – to keep it balanced, you need to add or subtract the same amount from both sides. This ensures that the inequality remains true. It’s important to stay organized and keep track of your calculations. Writing down each step helps prevent errors and makes it easier to follow your logic. Always remember to check your work; a quick review can catch any accidental slips. And stay focused, each tiny step gets you closer to the final solution. The key here is balance - whatever you do, do it to both sides to stay on track.

Step 3: Solve for x

Almost there! Now, we need to get $\bf{x}$ completely alone. Currently, it's being multiplied by $-2$. To undo this, we'll divide both sides of the inequality by $-2$. Important note: When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is a crucial rule to remember!

So, dividing both sides by $-2$, we get:

$\bf{\frac{3}{-2} \geq \frac{-2x}{-2}}$

This simplifies to:

$\bf{-1.5 \geq x}$

Or, if we write it with $x$ on the left side (which is more common):

$\bf{x \leq -1.5}$

And there you have it! We've solved for $x$. This step is the culmination of all our previous work. It's where we finally isolate the variable and find its possible values. Remember that rule about flipping the inequality sign – it's vital. Dividing by a negative number changes the direction of the inequality, ensuring the solution is accurate. Think about it this way: when you multiply or divide by a negative number, you're essentially reflecting the inequality across the number line. This changes which values satisfy the inequality. Always double-check your sign changes and calculations to avoid errors. Taking the time to be extra careful can make a huge difference in achieving the correct answer. The final solution is expressed, in this case, by all numbers less than or equal to -1.5. This means that any number smaller than or equal to -1.5 will make the original inequality true. Congrats, you made it!

Step 4: Verification

Let's verify that we've done everything correctly. Pick a number that satisfies our final inequality $x \leq -1.5$. Let's use $-2$. Now, let's plug $-2$ back into the original inequality:

$\bf{9.5 \leq -2((-2) - 3.25)}$

$\bf{9.5 \leq -2(-5.25)}$

$\bf{9.5 \leq 10.5}$

This is true! Since our choice of $-2$ made the original inequality true, we can be confident that our solution is correct. Verification is a great habit to develop. It's like a final check to ensure we didn't make any mistakes along the way. This step is a critical aspect, and can prevent errors if a mistake was made. Using a number within the range of your solution, helps to ensure that your final answer is correct. Choosing a number helps test the result. If the result turns out to be false, then you know there is some area that needs to be revisited, or corrected. This is a great habit to have and use, to ensure that the answer is always correct. Let's test a number that does not satisfy the final inequality, to verify that it does not work. Let's use 0.

$\bf{9.5 \leq -2((0) - 3.25)}$

$\bf{9.5 \leq -2(-3.25)}$

$\bf{9.5 \leq 6.5}$

This is false! This verifies that our solution, $x \leq -1.5$ is correct. It's a great habit to always verify your results.

Conclusion

And there you have it! We've successfully solved the inequality $9.5 \leq -2(x - 3.25)$. We followed these steps:

1. Distributed the $-2$.
2. Isolated the variable term.
3. Solved for $x$, remembering to flip the inequality sign when dividing by a negative number.
4. Verified our solution.

Solving inequalities may seem complicated, but with practice, it becomes second nature. Keep practicing, and you'll become a pro in no time! Remember to always check your work and stay organized. With consistent effort, you'll find yourself solving inequalities with ease. Keep up the great work, you guys! Keep in mind, this is just one example. You can practice with other problems, and learn a lot more from examples! Every problem will help you gain a better understanding, and help build your skills to do it on your own. Now go out there and solve some inequalities! You got this!