Solving Inequalities: A Step-by-Step Guide

Hey guys! Let's dive into the world of inequalities and figure out how to solve them, specifically focusing on the problem: $-10 \leq 3x - 1 \leq 11$. Don't worry; it's not as scary as it looks! We'll break it down into easy-to-follow steps, so you can ace these problems. Understanding how to solve inequalities is super important in math, so let's get started. This is the core concept, so pay attention, as the understanding of inequalities is fundamental to advanced math concepts. We're dealing with a compound inequality here, which means we have two inequalities rolled into one. Think of it as a double-sided problem; we need to work on both sides simultaneously to isolate the variable 'x'. The goal, as always, is to get 'x' by itself in the middle. Before you begin, make sure you fully understand the rules of inequalities. Remember that whatever you do to one part of the inequality, you MUST do to all parts to keep it balanced. This is exactly like solving an equation, except that we are dealing with ranges of values instead of a single value, so every step must be done to all three parts of the inequality. Let's begin! This method will work for any complex inequality that is of the same format as this one. The concept is the same; the values will be different.

Step 1: Isolate the Term with 'x'

Alright, our first step is to get the term with 'x' (which is 3x) by itself in the middle. Currently, we have -1 next to it. To get rid of that -1, we need to add 1 to all parts of the inequality. Remember, we need to do it to all three sections to keep everything balanced. So, let's do it:

-10 + 1 \leq 3x - 1 + 1 \leq 11 + 1

This simplifies to:

-9 \leq 3x \leq 12

See? We're already making progress! By adding 1 to each part, we've isolated the term containing 'x', bringing us one step closer to our goal of finding out the possible values of 'x'. Take a moment to understand this step because it's the foundation upon which we build the solution. Now you understand the importance of balancing both sides when solving inequalities. Remember, to do exactly the same thing to all three parts.

Step 2: Solve for 'x'

Now that we have -9 \leq 3x \leq 12, our next move is to isolate 'x' completely. Currently, 'x' is being multiplied by 3. To undo this multiplication, we need to divide all parts of the inequality by 3. So, let's do that:

\frac{-9}{3} \leq \frac{3x}{3} \leq \frac{12}{3}

This simplifies to:

-3 \leq x \leq 4

And there you have it! We've solved for 'x'. This tells us that 'x' can be any number greater than or equal to -3 and less than or equal to 4. This is called the solution set. In the solution set of -3 \leq x \leq 4, the 'x' values are contained between -3 and 4. In the process of solving the equation, it is important to pay close attention to the signs of the number.

Step 3: Understanding the Solution

So, what does -3 \leq x \leq 4 actually mean? It means that 'x' can be any value within this range, including -3 and 4. We can represent this solution on a number line. You'd draw a closed circle (because the inequality includes 'equal to') at -3 and 4 and shade the line in between, representing all the possible values for 'x'. It is useful to visualize the solution with the help of a number line to represent all the possible values that 'x' can take. The range is indicated on the number line. Any number between -3 and 4 will be the solution, including -3 and 4. Understanding the notation of the solution set helps us to interpret the results of the inequality and express the solutions clearly. It provides a concise way to describe the range of values that satisfy the inequality. You can further explore this result by plotting it on a graph. The ability to visualize the solution set will enable you to better understand the possible values that 'x' can take.

Step 4: Verification of the Solution

Let's check our work, guys. To verify, we can pick a value within the solution set and plug it back into the original inequality. Let's choose x = 0 (because it's easy!).

Substituting x = 0 into -10 \leq 3x - 1 \leq 11, we get:

-10 \leq 3(0) - 1 \leq 11

-10 \leq -1 \leq 11

This is true! So, our solution seems to be correct. Now, try it with another value, like x = 2. Substitute x = 2 into the original inequality:

-10 \leq 3(2) - 1 \leq 11

-10 \leq 5 \leq 11

This is also true, confirming our answer. The process of verifying the solution with values helps to ensure the accuracy of the answer. You can try different values within the range to test them.

Step 5: Expressing the Solution in Different Forms

We've already seen the solution in inequality form (-3 \leq x \leq 4). But there are other ways to express it. You can express the result in interval notation as well. This is another way to represent the solution. In interval notation, we use brackets [ ] to include the end values (since our inequality includes 'equal to') and parentheses ( ) to exclude them. So, our solution, -3 \leq x \leq 4, can be written in interval notation as [-3, 4]. This represents all real numbers from -3 to 4, including -3 and 4. This is just a different way of presenting the same solution. Familiarizing yourself with different notations helps improve your math skills.

Step 6: Addressing Common Mistakes

One of the most common mistakes is forgetting to apply the same operation to all parts of the inequality. Always remember to treat all three sections equally. Another mistake is not paying attention to the direction of the inequality signs when multiplying or dividing by a negative number (we didn't encounter this in our example, but it's crucial). It is also a common mistake to get confused while solving an inequality and an equation. Make sure you are always paying attention to your steps and the rules of the math.

Step 7: Practice Makes Perfect!

Solving inequalities becomes easier with practice. Try different examples, and don't hesitate to ask for help if you get stuck. Here's a practice problem for you to try: Solve for x: -5 < 2x + 3 < 7. Give it a shot, and see if you can apply the steps we've gone through. The more you practice, the better you'll become at solving inequalities. And that’s all there is to it, guys! We've successfully solved the inequality and discussed the different ways to represent the solution. Keep practicing, and you'll become a pro in no time. Always be confident when solving the inequality, and you will definitely get it right. Keep learning! I hope this guide helps. Feel free to ask me for more examples! Remember, practice is the key to success! Good luck, and happy solving!